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APPLETONS' 


CYCLOPEDIA  OF  TECHNICAL 

DRAWING. 


EMBRACING 


Jjrinnplts  of 

AS   APPLIED    TO 

PRACTICAL   DESIGN. 

WITH    NUMEROUS    ILLUSTRATIONS    OF 

TOPOGRAPHICAL,  MECHANICAL,   ENGINEERING, 

ARCHITECTURAL,   PERSPECTIVE,   AND 

FREE-HAND  DRAWING. 


EDITED  BY 

W.    E.    WORTHEN,    C.  E. 


NEW   YORK: 
D.     APPLETON    AND     COMPANY, 

1,  3,  AND  5  BOND   STREET. 
1892. 


COPYRIGHT,  1885, 
BY  D.  APPLETON  AND  COMPANY. 


PREFACE. 


"  AT  the  suggestion  of  the  publishers,  this  work  was  undertaken  to  form 
one  of  their  series  of  dictionaries  and  cyclopedias.  In  this  view,  it  has  been 
the  intention  to  make  it  a  complete  course  of  instruction  and  book  of  refer- 
ence to  the  mechanic,  architect,  and  engineer.  It  has  not,  therefore,  'been 
confined  to  the  explanation  and  illustration  of  the  methods  of  projection,  and 
the  delineation  of  objects  which  might  serve  as  copies  to  the  draughtsman, 
matters  of  essential  importance  for  the  correct  and  intelligible  representation 
of  every  form ;  but  it  contains  the  means  of  determining  the  amount  and 
direction  of  strains  to  which  different  parts  of  a  machine  or  structure  may  be 
subjected,  and  the  rules  for  disposing  and  proportioning  of  the  material  em- 
ployed, to  the  safe  and  permanent  resistance  of  those  strains,  with  practical 
applications  of  the  same.  Thus,  while  it  supplies  numerous  illustrations  in 
every  department  for  the  mere  copyist,  it  also  affords  suggestions  and  aids 
to  the  mechanic  in  the  execution  of  new  designs.  And,  although  the  arrang- 
ing and  properly  proportioning  alone  of  material  in  a  suitable  direction,  and 
adequately  to  the  resistance  of  the  strains  to  which  it  might  be  exposed, 
would  produce  a  structure  sufficient  in  point  of  strength  for  the  purposes  for 
which  it  is  intended,  yet,  as  in  many  cases  the  disposition  of  the  material 
may  be  applied  not  only  practically,  but  also  artistically,  and  adapted  to  the 
reception  of  ornament,  under  the  head  of  Architectural  Drawing,  the  general 
characteristics  of  various  styles  have  been  treated  of  and  illustrated,  with 
brief  remarks  on  proportion  and  the  application  of  color."  .  .  .  1857. 

Since  its  first  publication,  this  work  has  been  subjected  from  time  to  time 
to  revision.  It  has  now  been  deemed  necessary  to  almost  entirely  rearrange 
and  rewrite  it ;  to  add  largely  to  the  subject-matter  and  to  the  illustrations, 
introducing  examples  of  later  practice  and  experience ;  to  extend  the  scope 
of  the  work,  and  make  it  more  nearly  a  cyclopaedia  of  drawing  and  design. 
There  are  no  changes  in  the  principles  of  projection  as  applied  to  drawing, 
and  no  marked  improvement  in  drawing-instruments ;  but  in  the  present 
practice  finished  drawings  in  shade  and  color  are  exceptional.  It  is  suffi- 
cient, for  almost  every  purpose,  for  the  draughtsman  to  make  accurate  projec- 
tions with  pencil  on  paper,  and  trace  them  afterward  on  cloth.  The  pencil- 
drawings  can  be  readily  altered  or  amended,  and,  where  there  are  many  repe- 


iv  PREFACE. 

titions  of  the  same  parts,  but  a  single  one  may  be  drawn.  On  the  tracing 
they  are  made  complete,  and  these  are  preserved  as  originals  in  the  office, 
while  sun-prints  of  them  are  used  for  details  of  construction  in  the  shop,  or 
distributed  as  circulars  among  customers. 

In  the  sale  of  former  editions  of  this  work,  it  has  been  found  that  its 
success  has  been  largely  due  to  its  value  as  a  book  of  design.  Great  attention 
has  therefore  been  given  to  secure  practical  illustration  of  constructions  and 
machines  of  recent  date ;  the  nature  of  materials  in  common  use  has  been 
more  extensively  treated,  and  the  character  and  effect  of  stresses  and  strains, 
their  kind  and  direction,  more  fully  explained,  with  as  simple  rules  as  possible 
to  determine  them  for  practical  application. 

Of  late  years  the  science  of  "  graphics  "  has  become  of  great  importance, 
and  is  here  fully  illustrated  in  its  varied  applications,  showing  not  only  this 
method  of  recording  the  facts  of  the  statistician,  and  affording  comparisons 
of  circumstances  and  times,  the  growth  of  population,  the  quantities  and  cost 
of  agricultural  and  mechanical  production,  and  of  their  transport,  movements 
of  trade,  fluctuations  of  value,  the  atmospheric  conditions,  death-rates,  etc., 
but  also  in  its  application  to  the  plotting  of  formulae  for  their  ready  solution, 
by  the  draughtsman  and  designer.  For  many  of  the  rules  in  this  work  the 
results  of  the  formulae  of  various  authors  have  been  plotted  graphically,  and 
the  rule  given  one  deemed  of  the  greatest  weight,  not  always  by  average, 
but  most  consistent  with  our  own  experience. 

In  astronomical  calculations  every  decimal  may  have  its  importance.  It 
is  not  so  in  those  of  the  mechanical  or  architectural  designer ;  solutions  by 
graphics  are  sufficient  for  their  purpose,  and,  simpler  than  mathematical  cal- 
culations, they  are  thus  less  liable  to  error ;  it  is  very  good  practice  to  use  one 
as  a  check  on  the  other.  It  is  to  be  remarked  that  inaccuracy  in  facts,  and 
carelessness  in  observation,  are  not  eliminated  from  an  equation  by  closeness 
of  calculation,  and  when  factors  are  not  established  within  the  limits  of  units 
it  is  useless  to  extend  the  results  to  many  places  of  decimals.  It  is  of  the 
utmost  importance  to  know  at  first  well  the  object  and  purposes  of  the 
design,  the  stresses  to  which  its  parts  are  to  be  subjected,  and  the  strength 
and  endurance  of  the  materials  of  which  it  is  to  be  composed.  In  establish- 
ing rules  for  ourselves,  be  sure  of  the  facts,  and  that  there  are  enough  of 
them  for  a  general  application.  Rules  are  necessary,  but  their  application 
and  usefulness  depend  largely  on  the  experience  of  the  user,  and  life  must 
be  a  record  of  applications  and  effects.  It  is  comparatively  easy  to  make 
a  work  strong  enough ;  but  to  unite  economy  with  proportion  is  difficult. 

To  make  the  work  complete  in  itself,  so  as  to  form  a  sort  of  single  book 
for  most  of  the  purposes  of  the  draughtsman  and  designer — embracing  the 
profession  of  surveyor,  engineer,  and  architect — tables  of  logarithms,  latitudes 
and  departures,  squares  and  cubes,  and  square  and  cube  roots,  weights  and 
measures,  and  weights  of  material,  have  been  added. 

w. 


CONTENTS. 


PAGES 

CONSTRUCTION  OF  GEOMETRICAL  PROBLEMS 1-39 

Drawing  of  lines — straight,  curved,  and  parallels,  angles,  perpendicular ;  bisecting 
angles;  arcs  and  circles,  15.  On  polygons  and  circles;  triangles,  parallelograms, 
squares  ;  circles,  angles  ;  polygons  ;  inscribed  and  described  circles ;  pentagons,  hexa- 
gons, octagons ;  table  of  polygonal  angles,  23.  On  the  ellipse,  parabola,  hyperbola, 
cycloid,  epicycloid,  involute  and  spiral,  33.  Use  of  triangle  and  square,  33.  Areas 
of  figures,  37.  To  draw  squares  of  given  proportionate  sizes,  39. 

DRAWING  INSTRUMENTS 40-77 

Description  and  use  ;  rulers  ;  triangles;  T-square;  parallel  ruler  ;  sweeps  and  vari- 
able curves  ;  drawing  pens ;  dotting  point ;  pricking  point ;  compasses  ;  dividers ; 
beam  compasses  ;  porportional  dividers  ;  scales ;  scale  guard ;  diagonal  scales ;  ver- 
nier scales  ;  sector  ;  protractors  ;  pantagraphs  ;  camera  lucida  ;  drawing  table  and 
board,  56.  Drawing  paper ;  tracing  paper ;  tracing  cloth ;  mouth  glue ;  damp  stretch- 
ing paper ;  mounting  paper  and  drawings,  59.  Management  of  the  instruments ; 
ink  ;  exercises  with  drawing  pen  ;  various  letters  and  numerals  ;  cross-section  paper  ; 
diagrams  showing  use  of  cross-section  paper,  77. 

ORTHOGRAPHIC  PROJECTION 78-109 

Definitions;  points;  straight  line;  solid,  81.  Simple  bodies;  pyramid;  prism, 
87.  Construction  of  the  conic  sections,  90.  Penetration  or  intersection  of  solids ; 
cylinders,  cone,  and  sphere  ;  cylinder  and  ring  ;  sphere  and  prism  ;  cone  and  prism  ; 
cone  and  cylinder,  102.  Of  the  helix,  104.  Development  of  surfaces ;  cylinder ; 
cone;  sphere,  107.  Shade-lines,  109. 

SHADES  AND  SHADOWS 110-136 

Of  a  point :  straight  line  ;  solid  ;  circle ;  pyramid ;  cylinder  ;  cone  ;  shadows  cast 
upon  a  cylinder  by  various-shaped  caps  ;  shadows  cast  upon  a  prism ;  shadows  upon 
the  interior  of  a  cylinder,  hollow  hemisphere,  a  niche,  a  sphere  ;  line  of  shade  on  the 
surface  of  a  ring,  grooved  pulley,  square-threaded  nut  and  screw,  triangular-threaded 
nut  and  screw,  126.  Manipulation  of  shading  and  shadows — methods  of  tinting; 
surfaces  in  the  light ;  surfaces  in  shade  ;  shading  by  flat  tints  ;  by  softened  tints, 
129.  Elaboration  of  shading  and  shadows;  examples  of  finished  shading;  on  con- 
cave surfaces,  spheres,  ring,  cone,  flat  surfaces ;  colors  for  tints  ;  expeditious  way  of 
shading  a  cylinder ;  body  color ;  margin  of  light ;  washing  ;  conventional  tints  for 
materials,  136. 

PLOTTING ; 137-148 

Scales  ;  scales  prescribed  by  different  commissions,  138.  Variation  of  compass, 
1 39.  Plotting  compass  surveys  ;  balancing  error  ;  plotting  by  latitudes  and  depart- 
ures ;  area  by  latitudes  and  departures  ;  area  by  triangles  ;  plotting  by  offsets,  147. 
System  of  division  of  United  States  land,  148. 


VI 


CONTENTS. 


PAGES 

TOPOGRAPHICAL  DRAWING 149-180 

Conventional  signs ;  representation  of  hills  ;  contour  lines,  156.  Railway  surveys  ; 
profiles;  sections,  land  plans,  159.  Hydrometrical  or  marine  surveys,  conventionali- 
ties, 160.  Geological  and  statistical  features,  162.  Transferring  ;  pricking  through; 
by  tracing;  blue-print  process;  copying-glass;  transfer-paper;  pantagraph,  165. 
Map  projections ;  perspective  projection  on  planes ;  developed  perspective  projec- 
tions ;  projections  by  developing  elements  ;  projections  conforming  to  some  arbi- 
trary condition ;  polyconic  adopted  by  United  States  Coast  Survey ;  De  Lorgne's 
projection;  M creator's  chart,  171.  Colored  topography  ;  conventional  colors  ;  direc- 
tions; finishing;  lettering;  titles,  180. 

MATERIALS 181-199 

Earth  and  rocks,  182.  Building  materials  ;  wood,  185.  Stones;  technical  terms 
masonry;  granitic  stones,  argillaceous  stones  ;  sandstones ;  limestone,  188.  Artificial 
building  material ;  bricks  ;  tile  ;  terra-cotta  ;  mortars  ;  limes  ;  cement ;  concrete ; 
plastering,  191.  Metals;  conventional  hatchings ;  iron;  steel;  other  metals;  specific 
gravity ;  weight ;  melting-point ;  resistance  to  crushing  and  tension  ;  results  of  Prof. 
Thurston's  tests  of  metals,  196.  Sulphur;  glass;  rubber;  paints;  coals,  199. 

MECHANICS 200-219 

Force ;  center  of  gravity  ;  levers ;  wheel  and  axle ;  pulley ;  inclined  plane ;  wedge  ; 
screw  ;  inclined  forces  ;  parallelogram  of  forces  ;  hydraulic  press ;  velocity  of  falling 
bodies;  friction,  212.  Mechanical  work  or  effect;  horse-power,  etc.;  water-power; 
wind ;  steam  ;  steam  worked  expansively  ;  cut-offs  ;  compound  engines ;  indicator 
cards,  219. 

MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS 220-361 

Stress  ;  dead  load  ;  strength  of  posts  and  columns ;  Phoenix  columns,  etc. ;  shear- 

«  ing  stresses ;  torsional  stress ;  transverse  stress  ;  strength  of  beams  ;  tables  of  dimen- 
sions of  channel  beams  and  angle-iron  ;  composite  beams  ;  bolts  and  nuts  ;  strength 
of  bolts  ;  washers  ;  shafts  and  axles  ;  journals  ;  keys  ;  car-axles  ;  shafting  ;  bear- 
ings ;  couplings;  clutch;  pulleys;  belts;  ropes,  275.  Gearing;  epicycloidal  teeth; 
projections  of  a  spur-wheel  and  bevel-wheel,  294.  Drawing  of  screws,  297.  Fric- 
tional  gearing,  299.  Ropes  and  chains  ;  hooks ;  levers  ;  cranks ;  connecting-rods  ; 
steam-engine ;  working-beam ;  parallel-motion  links ;  steam-cylinders  and  pistons, 
331.  Valves  ;  hydrants,  342.  Riveted  joints  for  boilers  ;  boilers,  351.  Wrought- 
iron  pipe  connections,  355.  Frames  ;  governors  ;  fly-wheels  ;  air-chambers,  361. 

ENGINEERING  DRAWING 362-460 

Foundations ;  sheet-piling ;  retaining-walls ;  foundations  for  piei^s,  etc.,  375. 
Dams  ;  locks  of  canals  ;  conduits  ;  reservoirs,  395.  Water-pipes,  398.  Sewers,  401. 
Gas-supply,  402.  Roads,  407.  Roofs  and  bridges;  piers,  432.  Arch-bridges;  sus- 
pension-bridges, 438.  Boiler-setting ;  chimneys,  444.  Location  of  machines ;  ma- 
chine foundations,  449.  Tunnels,  453.  Railway  stock,  458.  Wave-line  principle  of 
ship  construction,  460. 


ARCHITECTURAL  DRAWING 

Details  of  construction  ;  concrete  walls,  468.  Frames  and  floors ;  fire-resisting 
floors,  474.  Groined  ceilings,  476.  Doors  ;  windows ;  moldings,  485.  Stairs,  492. 
Fireplaces  ;  flues  ;  roofs  ;  gutters  ;  plastering,  495.  Proportions  and  distribution  of 
rooms  and  passages,  500.  Plans  and  elevations  of  buildings  ;  stores  and  warehouses, 
521.  School-houses,  531.  Churches,  theatres,  lecture-rooms,  music  and  legislative 
halls ;  hospitals,  542.  Stables ;  cow-houses ;  greenhouses,  547.  Ventilation  and 
warming,  555.  Plumbing,  564.  Greek  and  Roman  orders  of  architecture,  596.  Or- 
naments of  the  Renaissance  ;  principles  of  design,  601. 


461-601 


CONTENTS. 


vn 


PAGES 

PERSPECTIVE  DRAWING 602-624 

Angular  perspective,  610.     Parallel  perspective,  624. 

ISOMETRICAL    DRAWING 625-638 


FREE-HAND  DRAWING 

Geometrical  figures  and  design,  643.  Proportions  of  the  human  frame,  647. 
Figure  drawing,  650.  Forms  of  animals,  653.  Illustrations  from  different  artists, 
664. 

APPENDIX 

Extracts  from  New  York  building  laws,  670.  Patent-Office  drawings,  670.  Men- 
suration ;  properties  of  triangles,  672.  Lineal  measure,  672.  Table  of  inches  in 
decimals  of  a  foot,  673.  Table  of  measures  of  surface,  673.  Table  of  measures  of 
capacity ;  dry  measure  ;  weights  ;  cubic  measure,  674.  Table  of  weight  of  rolled 
iron,  675.  Table  of  weight  of  wrought-iron  and  brass  plates  and  wire,  676.  Table 
of  weight  of  wrought-iron  welded  tubes  ;  boiler  tubes  ;  driven-well  tubes ;  heavy 
wrought  galvanized  iron  spiral  riveted  pipes,  678.  Table  of  copper  and  brass  rods, 
678.  Table  of  number  of  Burden's  rivets  in  100  pounds,  679.  Table  of  number  of 
wrought  spikes  to  a  keg,  679.  Table  of  length  of  cut  nails  and  spikes,  and  number 
in  a  pound,  680.  Table  of  weights  of  lead  pipe  per  foot,  680.  Table  of  the  weight 
of  a  cubic  foot  of  water  at  different  temperature?,  680.  Table  of  properties  of  satu- 
rated steam,  682.  Table  of  mean  pressures  in  steam  cylinders  at  different  rates  of 
expansion,  683.  The  flow  of  water,  with  table  of  discharge  over  weir,  685.  Flow 
of  water  through  pipes  and  sewers,  689.  Flow  of  air  through  pipes,  690.  Table  of 
circumferences  and  areas  of  circles,  695.  Table  of  squares,  cubes,  square  and  cube 
roots  of  numbers,  703.  Table  of  latitudes  and  departures,  709.  Table  of  natural 
sines  and  cosines,  718.  Logarithms  of  numbers,  735. 

INDEX. 

DESCRIPTION  OF  PLATES. 

PLATES  I  TO  XIV. 

SCRAPS. 


639-664 


665-735 


DESCRIPTION  OF  PLATES. 


PLATE 

I.   Shading  of  prism  and  cylinder  by  flat  tints.     Referred  to  on  pages 

126-7. 

II.  Shading  of  cylinder  and  segment  of  hexagonal  pyramid.  Referred  to 
on  pages  128-9. 

Ill,  IV.  Finished  shading  and  shadows  of  different  solids.      Referred  to  on 
page  131. 

V.  Shades  and  shadows  on  screws.     Referred  to  on  page  126. 
VI.  Example  of  topographical  drawing,  done  entirely  with  the  pen. 
VII.  The  same,  with  the  brush,  in  black. 
VIII.  The  same,  with  the  brush,  in  color.     Referred  to  on  page  174. 

IX.  Contoured  map  of  Staten  Island,  shaded  by  superimposed  washes, 
the  washes  increasing  in  intensity  or  strength  as  required  to  pro- 
duce the  effect. 

X.  Geological  map  of  part  of  New  Jersey,  colored  to  show  the  different 
formations. 

XI.  Finished,  shaded  sectional  view,  colored  to  show  the  different  metals, 
of  a  balanced  leather  cup-valve.  The  body  is  of  cast-iron  ;  the 
piston,  brass  ;  packing,  leather  ;  piston-rod,  wrought-iron — this 
last  not  distinctively  colored. 

XII.  Finished  perspective   drawing,  with  shades  and  shadows,  of  a  large 
bevel-wheel  and  two  pinions,  with  shifting  clutches. 

XIII.  Front  elevation  of  a  building,  in  color. 

XIV.  Perspective  view  of  Gothic  church,  finished  in  color. 

XV.  Plan,  elevation,  and  section  of  bevel-wheel,  pinion,  and  clutches, 
shown  in  perspective  Plate  XII. 

XVI  to  XX.  Details  of  progressive  perspective  projections  of  Plate  XV,  as 
shown  completed  in  Plate  Xll. 


APPLETONS' 


CYCLOPAEDIA  OF  DRAWING, 


CONSTRUCTION  OF   GEOMETRICAL  PROBLEMS. 


MOST  persons,  at  some  time,  have  made  use  of  the  simple  drawing  instru- 
ments, pencils,  straight-edges  or  rulers,  and  compasses  or  dividers  with  change- 
able points,  and  many  suppose  that  there  can  be  no  skill  in  their  use  ;  but  to 
one  critical  in  these  matters  there  are  great  differences  to  be  observed  even  in 
common  drawings,  in  the  straightness  and  uniformity  of  the  lines,  and  in  the 
care  of  the  surface  of  the  paper. 

Select  for  the  geometrical  problems  and 
for  usual  drawings  a  No.  3  or  H  H  H  pen- 
cil. It  should  be  sharpened  to  a  cone-point 
(Fig.  1).  Where  a  pencil  is  used  for  drawing 
lines  only,  some  draughtsmen  sharpen  the 
pencil  to  a  wide  edge,  like  a  chisel. 

In  drawing  a  straight  line,  hold  the  ruler 
firmly  with  the  left  hand  ;   with  the  right 
hand   hold   the  pencil   lightly  but   without 
FlG-  1-  slackness,  and  a  little  inclined  in  the  direc- 

tion of  the  line  to  be  drawn,  keeping  the  pencil  against  the  edge  of  the 
ruler,  and  in  the  same  relative  position  to  the  edge  during  the  whole  operation, 
of  drawing  the  line. 


2  CONSTRUCTION  OF  GEOMETRICAL  PROBLEMS. 

To  draw  a  clean  line  and  preserve  the  point  of  the  pencil,  the  part  of  the 
cone  a  little  above  the  point  of  the  pencil  should  bear  against  the  edge  of  the 
ruler,  and  the  pencil  should  be  carried  steadily  while  drawing.  Any  oscilla- 
tion will  throw  the  point  farther  from  or  nearer  the  ruler,  and  the  line  will 
not  be  straight  (Fig.  2). 


FIG.  2. 

In  the  use  of  the  compasses  do  not  make  a  hole  through  the  paper  with  the 
needle  or  sharp  point,  but  only  into  the  paper  sufficient  to  maintain  the 
position. 

Keep  the  paper  clean,  and  use  rubber  as  little  as  possible. 

As  drawing  is  based  on  geometrical  principles,  we  commence  with  geo- 
metrical definitions  and  problems  to  give  the  learner  some  knowledge  of 
the  science  of  geometry,  with  a  valuable  exercise  in  the  use  of  drawing 
instruments. 

Geometrically  a  point  is  defined  as  position  merely  :  in  drawing,  the  posi- 
tions of  points  are  marked  on  the  paper  by  prick-marks  of  a  needle  or  sharp 
point,  and  by  the  dot  of  a  pencil ;  sometimes  inclosed  O,  sometimes  designated 
by  the  intersection  of  two  short  lines  X  >. 

Geometrically  lines  have  but  one  dimension,  .length,  and  the  direction 
of  a  line  is  the  direction  from  point  to  point  of  the  points  of  which  the 
line  is  composed  :  in  drawing,  lines  are  visible  marks  of  pencil  or  pen  upon 
paper. 


FIG.  3. 

A  straight  line  is  such  as  can  be  drawn  along  the  edge  of  the  ruler,  and  is 
one  in  which  the  direction  is  the  same  throughout.  In  drawing  a  straight  line 
through  two  given  points,  place  the  edge  of  the  ruler  very  near  to  and  at  equal 
distances  from  the  points,  as  the  point  of  the  pencil  or  pen  should  not  be  in 
contact  with  the  edge  of  the  ruler  (Fig.  3). 

Lines  in  geometry  and  drawing  are  generally  of  limited  extent.     A  given 


CONSTRUCTION  OF  GEOMETRICAL  PROBLEMS. 


3 


FIG.  4. 


or  known  line  is  one  established  on  paper  or  fixed  by  dimensions.     Lines  of 
the  same  length  are  equal. 

To  draw  Curved  Lines. — Insert  the  pencil-point  in  the  compasses,  and  open 
them  to  a  suitable  extent.  With  the  needle  or  sharp  point  resting  on  the 
paper  describe  a  line  with  the  pencil  around  this  point ;  the  line  thus 
described  is  usually  called  a  circle — more  strictly  it  is  the  circumference  of  a 
circle — the  circle  being  the  space  inclosed.  A  portion 
of  a  circumference  is  an  arc.  The  point  around 
which  the  circumference  is  described  is  the  center 
of  the  circle  (Fig.  4). 

If  a  line  be  drawn  from  the  center  to  the  circum- 
ference it  is  called  a  radius.  As  it  is  the  length 
embraced  between  the  points  of  the  compasses,  it  is 
often  called  by  mechanics  the  sweep. 

If  a  line  be  drawn  through  the  center,  and  limited 
by  the  circumference,  it  is  called  the  diameter,  and  is 
equal  to  two  radii. 

A  radius  is  a  semi-diameter  ;  a  diameter  is  the  longest  line  that  can  be  con- 
tained within  a  circumference.  Lines  limited  by  the  circumference,  and  which 
are  not  diameters,  are  chords. 

It  will  be  observed  that  arcs  are  lines  which  are  continually  changing  the 
directions,  and  are  called  curved  lines,  but  there  are  other  curved  lines  than 
those  described  by  compasses,  of  which  the  construction  will  be  explained 
hereafter. 

Besides  straight  and  curved  lines  there  are  often  lines,  in  drawing,  which 
can  neither  be  drawn  by  rulers  or  compasses,  as  lines  representing  the  direc- 
tions of  brooks  and  rivers,  the  margins  of  lakes  and  seas,  points  in  which  are 
established  by  surveys,  defined  on  paper,  and  connected  by  hand-drawing. 
These  may  be  called  irregular  or  crooked  lines. 

Where  it  is  necessary  to  distinguish  lines  by  names,  we  place  at  their 

extremities  letters  or  figures,  as  A B,  1 2  ;  the  line  A  B,  or  1  2. 

But  in  lines  other  than  straight,  or  of  considerable  extent,  it  is  often  necessary 
to  introduce  intermediate  letters  and  figures,  as  a  a  a. 


In  the  following  problems,  unless  otherwise  implied  or  designated,  where 
lines  are  mentioned,  straight  lines  are  intended. 

If  we  conceive  a  straight  line  to  move  sideways  in  a  single  direction,  it  will 
sweep  over  a  surface  which  is  called  a  plane.  All  drawings  are  projections  on 
planes  of  paper  or  board. 

Two  lines  drawn  on  paper,  and  having  the  same  direction,  can  never  come 
any  nearer  each  other,  and  must  always  be  at  the  same  distance  apart,  however 
far  extended.  Such  lines  are  called  parallel  lines. 


4:  CONSTRUCTION  OF  GEOMETRICAL  PROBLEMS. 

PEOB.  L — To  draw  a  line  parallel  to  a  given  line,  and  at  a  given  distance 
from  it  (Fig.  5). 

Draw  the  line  A  B  for  the  given  line,  and  take  in  the  compasses  the  dis- 
tance A  C — the  distance  at  which  the  other  line  is  to  be  drawn.  On  A,  as  a 


Jj 


FIG.  5. 


FIG.  6. 


center,  describe  an  arc,  and  on  B,  as  a  center,  another  arc  ;  draw  the  line  C  D 
just  touching  these  two  arcs,  which  will  be  the  parallel  line  required. 

PKOB.  II. — To  draw  a  line  parallel  to  a  given  line  through  a  given  point 
outside  this  line  (Fig.  6). 

Draw  the  given  line  A  B,  and  mark  the  given  point  C.  With  C  as  a  centei, 
find  an  arc  that  shall  only  just  touch  A  B ;  and  with  B  as  a  center,  and  the 
same  radius,  describe  an  arc  D.  Draw  through  the  point  C  a  line  just  touching 
this  last  arc,  and  the  line  C  D  will  be  the  parallel  line  required. 

Two  lines  in  the  same  plane,  not  parallel  to  each  other,  will  come  together 
if  extended  sufficiently  far.  The  coming  together,  cutting,  or  intersection  of 
two  lines,  is  called  an  angle  (Fig.  7). 

If  but  two  lines  come  together,  the  angle  may  be  designated  by  a  single 
letter  at  the  vertex,  as  the  angle  E. 

But,  if  three  or  more  lines  have  a  common  vertex,  the  angles  are  designated 
by  the  lines  of  which  they  are  composed,  as  the  angle  D  B  C  of  the  lines  D  B 

D 


FIG,  8 


and  B  C  ;  the  angle  A  B  C  of  A  B  and  B  C  ;  the  angle  A  B  D  of  A  B  and  B  D. 
The  letter  at  the  vertex  is  not  repeated,  and  must  always  be  the  central  letter. 
Describe  a  circle  (Fig.  8).  Draw  the  diameter  A  B.  From  A  and  B 
as  centers,  with  any  opening  of  the  compasses  greater  than  the  radius, 
describe  two  arcs  cutting  each  other  as  at  D.  Through  the  intersection 
of  these  arcs  and  the  center  C,  draw  the  line  D  E.  D  E  makes,  with  the 
diameter  A  B,  four  angles,  viz.,  A  C  D,  D  0  B,  B  C  E,  and  E  C  A.  Angles 


A* 

CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 


are  equal  whose  lines  have  equal  inclination  tfc^ach  other,  and  whose  lines,  if 
placed  one  on  the  other,  would  coincide.     By  construction,  the  points  C  and  D 


have,  respectively,  equal  distances  from  A  and  B  ;  the  line  D  C  can  not,  there- 
fore, be  inclined  more  to  one  side  than  to  the  other,  and  the  angle  A  C  D  must 
be  equal  to  the  angle  BCD.  Such  angles  are  called  right  angles.  It  can  be 
readily  proved  that  all  the  four  angles,  formed  by  the  intersection  of  D  E  with 
A  B,  are  equal,  and  are  right  angles. 

The  angles  A  C  D  and  D  C  B,  on  the  same  side  of  A  B,  are  called  adjacent 
angles  ;  as  also  DOB  and  B  C  E,  on  the  same  side  of  D  E. 

When  a  line,  standing  on  another  line,  makes  the  two  adjacent  angles  equal, 
the  angles  are  right  angles,  and  the  first  line  is  perpendicular  to  the  other. 

If  the  second  or  base  line  be  parallel  with  the  surface  of  still  water, 
it  is  called  an  horizontal  line,  and  the  perpendicular  line  is  called  a  ver- 
tical line. 

Draw  the  line  C  F.  It  will  be  observed  that  the  angle  F  C  D  is  less  than 
a  right  angle,  and  it  is  called  an  acute  angle  ;  the  angle  F  C  A  is  greater  than 
a  right  angle,  and  it  is  called  an  obtuse  angle.  It  will  be  observed  that,  no 
matter  how  many  lines  be  drawn  to  the  center,  the  sum  of  all  the  angles  on 
the  one  side  of  A  B  can  only  be  equal  to  two  right  angles,  and,  on  both  sides 
of  A  B,  can  only  be  equal  to  four  right  angles.  It  will  be  observed  that  the 
angles  at  the  center  include  greater  or  less  arcs  between  their  sides,  according 
to  the  greater  or  less  inclination  of  their  sides  to  each  other ;  that  the  right 
angles  intercept  equal  arcs,  and  that,  no  matter  how  large  the  circle,  the  pro- 
portion of  the  circle  intercepted  by  the  sides 
of  an  angle  is  always  the  same,  and  that  the 
arcs  can  therefore  be  taken  as  the  measures 
of  angles.  For  this  purpose  the  whole  cir- 
cumference is  supposed  to  be  divided  into 
three  hundred  and  sixty  degrees  (360°),  each 
degree  subdivided  into  sixty  minutes  (60'), 
and  each  minute  into  sixty  seconds  (60*). 
Each  right  angle  has  for  its  measure  one 
quarter  of  the  whole  circumference  (-^p-), 
or  90°,  and  is  called  a  quadrant. 

PROB.  III. — To  construct  an  angle  equal 
to  a  given  angle  (Fig.  9). 

Draw  any  angle,  as  C  A  B,  for  the  given 
angle,  and  the  line  a  b  as  the  base  of  the 
required  angle.  From  A,  with  any  suitable 
radius,  describe  the  arc  B  C,  and  from  a, 
with  the  same  radius,  describe  the  arc  b  c. 
Measure  the  length  of  the  arc  B  C,  or  rather 
the  chord,  that  is,  the  distance  in  a  straight  line  from  B  to  C,  and  lay  off  the 
same  distance  on  the  arc  b  c.  Draw  the  line  a  c,  and  the  angle  cab  will  be 
equal  to  C  A  B. 

PROB.  IV. — To  construct  an  angle  of  sixty  degrees  (Fig.  10). 

Lay  off  any  base-line,  and  from  A,  with  any  radius,  describe  an  arc,  and 


Fio.  9. 


6  CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 

from  B,  with  the  same  radius,  describe  another  arc  cutting  the  first,  as  at  C. 
Draw  the  line  C  A,  and  the  angle  CAB  will  be  an  angle  of  sixty  degrees. 
The  reason  of  this  construction  will  be  readily  understood  if,  on  the  cir- 


\ 


FIG.  10. 


;#» 
Fio.  11. 


cumference  of  any  circle,  chords  equal  to  the  radius  are  stepped  off  succes- 
sively. Six  will  exactly  complete  the  circle,  making  360°,  or  each  60°,  and 
the  angle  corresponding  will  be  60°. 

PKOB.  V. — To  draw  a  perpendicular  to  a  line  from  a  point  without  the 
line  (Fig.  11). 

Draw  a  line,  and  mark  the  given  point  outside  it,  A.  From  A  as  a  center, 
with  a  suitable  radius,  describe  an  arc  cutting  the  line  at  G  and  F.  From  G 
and  F,  as  centers,  describe  arcs  cutting  each  other.  The  line  drawn  through 
the  point  A,  and  the  point  of  intersection  E,  will  be  perpendicular  to  the 
line  G  F. 

The  radial  line  A  E  divides  the  chord  G  F  and  the  arc  G  E  F  into  two 
equal  parts  ;  and,  conversely,  the  line  perpendicular  to  the  middle  point  of  a 
chord  of  a  circle  is  radial — passes  through  the  center  of  that  circle. 

PROS.  VI. — To  draw  a  perpendicular  to  a  line  from  a  point  within  that 
line  (Fig.  12). 

1st  Method. — Draw  a  line,  and  take  the  point  A  in  the  line.  From  A,  as 
a  center,  describe  arcs  cutting  the  line  on  each  side  at  B  and  C.  From  B  and 


Nr 


£ 


,'D 


A 

FIG.  12. 


Cl 


Fia.  13. 


C,  as  centers,  describe  intersecting  arcs  at  D.     Draw  a  line  through  D  and  A> 
and  it  will  be  perpendicular  to  the  line  B  C  at  A. 


CONSTRUCTION  OF  GEOMETRICAL  PROBLEMS. 


\C/ 


2d  Method  (Fig.  13). — Draw  the  line,  and  mark  the  point  A  as  before. 
From  any  center  F,  without  the  line,  and  not  directly  over  A,  with  a  radius 
equal  to  F  A,  describe  more  than  a  half-circle  cutting  the  line,  as  at  D.  From 
D,  through  the  center  F,  draw  a  line  cutting  the  arc  at  E.  Draw  A  E,  and  it 
will  be  the  perpendicular  to  the  line  A  D. 

It  will  be  observed  that  the  line  D  E  is  the  diameter  of  a  circle,  and  that 
the  angle  DAE,  with  its  vertex  at  A  in  the  circumference,  would  embrace 
with  its  sides  half  a  circle,  had  a  full  circle  been  described.  It  has  been  shown 
that  angles  at  the  center  of  a  circle  have  for  their  measure  the  arc  embraced 
by  their  sides.  It  is  easily  demonstrable  that  angles,  with  their  vertices  in  the 
circumference,  have  for  their  measure  half  the  arc  embraced  by  their  sides, 
and,  consequently,  angles  embracing 
half  a  circumference  are  right  an- 
gles, and  their  sides  are  perpendicu- 
lar to  each  other. 

PROB.  VII. — To  draw  a  perpen- 
dicular to  the  middle  point  of  a  line 
(Fig.  14). 

From  the  extremities  A  and  B 
of  the  line,  as  centers,  describe  in- 
tersecting arcs  above  and  below  the 
line.  Through  these  intersections 
draw  the  line  0  D.  It  will  be  per- 
pendicular to  the  line  A  B,  and  bi- 
sect or  divide  it  into  two  equal  parts. 

If  the  line  A  B  be  considered  the  chord  of  a  circle,  its  center  would  lie  in 
the  line  C  D. 

This  construction  is  sometimes  used  merely  to  divide  a  line  into  two  equal 
parts,  or  bisect  it ;  but  if  we  have  dividers  or  compasses,  with  both  points 
sharp,  it  can  be  more  readily  done  with  them  (Fig.  15). 

Place  one  point  of  the  dividers  on  one  end  of  the  line,  and  open  the 
dividers  to  a  space  as  near  as  may  be  half  the  line.  Turn  the  dividers  on  the 
central  point ;  if  the  other  point  then  falls  exactly  on  the  opposite  extremity 

DIE 


FIG.  14. 


FIG.  15. 

of  the  line,  it  is  properly  divided  ;  but,  if  the  point  falls  either  within  or  with- 
out the  extremity  of  the  line,  divide  the  deficit  or  excess  by  the  eye,  in 
halves,  and  contract  or  extend  the  dividers  by  this  measure.  Then  apply  the 
dividers  as  before,  and  divide  deficit  or  excess  till  a  revolution  exactly  covers  the 
length  of  the  line.  By  accustoming  one's  self  to  this  process,  the  eye  is  made 
accurate,  and  one  estimate  is  sufficient  for  a  correct  division  of  any  deficit  or 


8 


CONSTRUCTION  OF   GEOMETRICAL  PROBLEMS. 


excess.  By  a  similar  process  it  is  evident  that  a  line  can  be  divided  into  any 
number  of  equal  parts,  by  assuming  an  opening  of  the  dividers  as  nearly  as 
possible  to  that  required  by  the  division,  and,  after  spacing  the  line,  dividing 
the  deficit  or  excess  by  the  required  number  of  parts,  contracting  or  expanding 
the  dividers  by  one  of  these  parts,  and  spacing  the  line  again,  and  so  on  till  it 
is  accurately  divided. 

PKOB.  VIII.  —  To  bisect  a  given  angle  (Fig.  16). 

Construct  an  angle,  and  from  its  vertex  A,  as  a  center,  describe  an  arc 
cutting  the  two  sides  of  the  angle  at  B  and  C.  From  B  and  C,  as  centers, 
describe  intersecting  arcs.  Draw  a  line  through  A  and  the  point  of  intersec- 
tion D,  and  this  line  will  bisect  the  angle. 

-B 


—6 


•I) 


FIG.  16. 


FIG.  17. 


PROB.  IX. — To  Used  an  angle  when  the  vertex  is  not  on  the  paper  (Fig.  17). 

Draw  two  lines,  A  B  and  E  C,  inclined  to  each  other,  but  not  intersecting. 
Draw  two  lines  intersecting  each  other,  a  b  and  a  c,  inside  and  parallel  to  A  B 
and  E  C.  Bisect  b  a  c  by  the  line  a  d,  and  this  line  will  also  bisect  the  angle 
whose  vertex  is  not  on  the  paper. 

PROB.  X. — Through  two  given  points  to  describe  an  arc  of  a  circle  with  a 
given  radius  (Fig.  18). 

From  B  and  C,  the  two  given  points,  with  an  opening  of  the  dividers  equal 
to  the  given  radius,  describe  two  arcs  crossing  at  A.  From  A,  as  a  center, 
with  the  same  radius,  describe  an  arc,  and  it  willbe  the  one  required. 


FIG.  18. 


FIG.  19. 


PROB.  XL — To  find  the  center  of  a  given  circle,  or  of  an  arc  of  a  circle. 

Of  a  circle  (Fig.  19). — Draw  the  chord  A  B.  Bisect  it  by  the  perpen- 
dicular C  D,  whose  extremities  lie  in  the  circumference,  and  bisect  C  D.  Gr, 
the  point  of  bisection,  will  be  the  center  of  the  circle. 


CONSTRUCTION   OF  GEOMETRICAL  PROBLEMS. 


Of  an  arc,  or  of  a  circumference  (Fig.  20). — Select  the  points  A,  B,  and  C 
in  the  circumference,  well  apart.  With  the  same  radius  from  A  and  B  as 
centers,  and  then  from  B  and  C  as  centers,  describe  arcs  cutting  each  other ; 
draw  the  two  lines  D  E  and  F  G  through  their  intersections.  The  point  0, 
where  these  lines  meet,  is  the  center  required. 

PKOB.  XII.  —  To  describe  a  circle  passing  through  three  given  points 
(Fig.  20). 

Proceed,  as  in  the  last  problem,  to  find  the  center  0.  From  0,  as  a 
center,  with  a  radius  0  A,  describe  a  circle,  and  it  will  be  the  one  required. 


FIG.  20. 

PKOB.   XIII.  —  To  describe  a  circle  passing  through  three  given 
where  the  center  is  not  available. 

1st  Method  (Fig.  21). — From  the  extreme  points  A  and  B,  as  centers, 
describe  the  arcs  B  G  and  A  H.  Through  the  third  point,  C,  draw  A  E  and 
B  F,  cutting  the  arcs.  Divide  the  arcs  A  F  and  B  E  into  any  number  of  equal 
parts,  and  set  off  a  series  of  equal  parts  of  the  same  length  on  the  upper  por- 
tions of  the  arcs  beyond  E  and  F.  Draw  straight  lines,  B  L,  B  M,  etc.,  to  the 
points  of  division  in  A  F,  and  A  I,  A  K,  etc.,  to  the  points  of  division  in  E  G ; 
the  successive  intersections  N,  0,  etc.,  of  these  lines  are  points  in  the  circle 
required  between  the  given  points  A  and  C,  which  may  be  filled  in  accord- 
ingly. Similarly,  the  remaining  part  of  the  curve,  B  C,  may  be  described. 

Zd  Method  (Fig.  22).— Let  A,  D,  and  B  be  the  given  points.  Draw  A  B, 
A  D,  and  D  B.  Draw  e  f  parallel  to  A  B.  Divide  D  A  into  a  number  of 
equal  parts  at  1,  2,  3,  etc., 
and  from  D  describe  arcs  "'• 
through  these  points  to  meet 
ef.  Divide  the  arc  A  e  into 
the  same  number  of  equal 
parts,  and  draw  straight 
lines  from  D  to  the  points 
of  division.  The  intersec- 
tions of  these  lines  successively  with  the  arcs  are  points  in  the  circle,  which 
may  be  filled  in  as  before. 

Note. — The  second  method  is  not  perfectly  true,  but  sufficiently  so  for  arcs 
less  than  one  fourth  of  a  circle. 


10 


CONSTRUCTION   OF  GEOMETRICAL  PROBLEMS. 


To  describe  the  arc  mechanically. — Let  a,  c,  I  be  the  three  points  of  a  curve  ; 
transfer  these  points  to  a  piece  of  stout  card-board,  and  draw  the  lines  a  c  and 
c  I,  and  extend  them  beyond  a  and  I.  Cut  out  the  card-board  along  these 


FIG.  23. 

lines.  Insert  upright  pins  on  the  points  a  and  I  of  the  drawing,  and  placing 
the  edges  of  the  cut  card-board  against  them,  and  maintaining  the  contact  of 
the  edges  of  the  card-board  with  the  pins,  slide  the  card  each  way.  Dot  the 
positions  of  the  vertex  of  the  angle  c,  and  the  dots  will  be  points  in  the  curve. 

PROB.  XIV. — To  draw  a  tangent  to  a  circle  from  a  given  point  in  the  cir- 
cumference. 

1st  Method  (Fig.  24). — Through  the  given  point  A  draw  the  radial  line 
A  C.  The  perpendicular  F  G-  to  that  line  will  be  the  tangent  required. 


FIG.  24. 


FIG.  25. 


2d  Method  (Fig.  25). — From  the  given  point  A  set  off  equal  arcs,  A  B  and 
A  D.  Join  B  and  D.  Through  A  draw  A  E  parallel  to  B  D,  and  it  will  be 
the  tangent  required.  This  method  is  useful  when  the  center  is  inaccessible. 

PROB.  XV. — To  draw  tangents  to  a  circle  from  a  point  without  it  (Fig.  26). 

From  the  given  point  A  draw  A  0  to  the  center  of  the  circle.    From  D,  the 


FIG.  26. 


FIG.  27. 


intersection  of  A  C  with  the  circle,  describe  an  arc,  with  a  radius  D  C,  cutting 
the  circle  at  E  and  F.    Draw  A  E  and  A  F,  and  they  will  be  the  tangents  required. 


CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 


11 


To  construct  within  the  sides  of  an  angle  a  circle  tangent  to  these  sides  at  a 
given  distance  from  the  vertex  (Fig.  27). — Let  a  and  b  be  the  given  points 
equally  distant  from  the  vertex  A.  Draw  a  perpendicular  to  A  C  at  a,  and  to 
A  B  at  I.  The  intersection  of  these  perpendiculars  will  be  the  center  of  the 
required  circle. 

In  the  same  figure,  to  find  the  center,  the  radius  being  given,  and  the 
points  a  and  b  not  known.  Draw  lines  parallel  to  A  C  and  A  B,  at  a  distance 
equal  to  the  given  radius,  and  their  intersection  will  be  the  center  required. 

PROB.  XVI. — To  describe  a  circle  from  a  given  point  to  touch  a  given 
circle  (Figs.  28  and  29). 

D  E  being  the  given  circle,  and  B  the  given  point,  draw  a  line  from  B  to 
the  center  C,  and  produce  it,  if  necessary,  to  cut  the  circle  at  A.  From  B, 


FIG.  28. 


FIG.  29. 


as  a  center,  with  a  radius  equal  to  B  A,  describe  the  circle  F  G,  touching  the 
given  circle,  and  it  will  be  the  circle  required. 

The  operation  is  the  same  whether  the  point  B  is  within  or  without  the 
circle. 

It  will  be  remarked  that,  in  all  cases  of  circles  tangent  to  each  other, 
their  centers  and  their  points  of  contact  must  lie  in  the  same  straight  line. 

PROB.  XVII. — To  draw  tangents  to  two  given  circles. 

1st  Method  (Fig.  30). — Draw  the  straight  line  ABC  through  the  centers 
of  the  two  given  circles.  From  the  centers  A  and  B  draw  parallel  radii,  A  D 


FIG.  30. 

and  B  E,  in  the  same  direction.  Draw  a  line  from  D  to  E,  and  produce  it  to 
meet  the  center  line  at  C  ;  and  from  C  draw  tangents  to  one  of  the  circles  by 
Problem  XV.  Those  tangents  will  touch  both  circles  as  required. 

2d  Method  (Fig.  31). — Draw  the  line  A  B  connecting  the  two  centers. 
Draw  in  the  larger  circle  any  radius,  A  H,  on  which  set  off  H  G,  equal  to  the 


12 


CONSTRUCTION   OF  GEOMETRICAL  PROBLEMS. 


radius  of  the  smaller  circle.     On  A  describe  a  circle  with  the  radius  A  G,  and 
draw  tangents,  B  I  and  B  K,  to  this  circle  from  the  other  center,  B.     From  A 


FIG.  81. 

and  B  draw  perpendiculars  to  these  tangents.     Join  C  and  D,  also  E  and  F. 
The  lines  0  D  and  E  F  will  be  the  required  tangents. 

Note. — The  second  method  is  useful  when  the  diameters  of  the  circles  are 
nearly  equal. 

PKOB.  XVIII.  —  Between  two  inclined  lines  to  draw  a  series  of  circles 
touching  these  lines  and  touching  each  other  (Fig.  32). 

Bisect  the  inclination  of  the  given  lines  A  B  and  C  D  by  the  line  N  0  ;  this 

is  the  center  line  of  the  circles  to  be 
inscribed.  From  a  point,  P,  in  this 
line,  draw  P  B  perpendicular  to  the 
line  A  B  ;  and  from  P  describe  the 
circle  B  D,  touching  the  given  lines, 
and  cutting  the  center  line  at  E. 
From  E  draw  E  F  perpendicular  to 
the  center  line,  cutting  A  B  at  F ; 
from  F  describe  an  arc,  with  a  ra- 
dius, F  E,  cutting  A  B  at  G ;  draw 
G  H  parallel  to  B  P,  giving  H  the 

center  of  the  second  touching  circle,  described  with  the  radius  H  E  or  II  G. 
By  a  similar  process  the  third  circle,  I  N,  is  described.     And  so  on. 

Inversely,  the  largest  circle  may  be  described  first,  and  the  smaller  ones  in 
succession. 

Note. — This  problem  is  of  frequent  use  in  scroll-work. 
PROB.  XIX. — Between  tivo  inclined  lines  to  draw  a  circular  arc  to  fill  up 
the  angle,  and  touching  the  lines  (Fig.  33). 

Let  A  B  and  D  E  be  the  inclined  lines.  Bisect  the  inclination  by  the  line 
F  C,  and  draw  the  perpendicular  A  F  D  to  define  the  limit  within  which  the 
circle  is  to  be  drawn.  Bisect  the  angles  A  and  D  by  lines  cutting  at  C, 
and  from  C,  with  radius  C  F,  draw  the  arc  H  F  G,  which  will  be  the  arc 
required. 

PROB.  XX. — To  fill  up  the  angle  of  a  straight  line  and  a  circle,  with  a  cir- 
cular arc  of  a  given  radius  (Fig.  34). 

On  the  center  C,  of  the  given  circle  A  D,  with  a  radius  C  E  equal  to  that 
of  the  given  circle  plus  that  of  the  required  arc,  describe  the  arc  E  F.  Draw 


FIG.  32. 


CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 


13 


G  F  parallel  to  the  given  line  H  I,  at  the  distance  G  H,  equal  to  the  radius 
of  the  required  arc,  and  cutting  the  arc  E  F  at  F.     Then  F  is  the  required 


H  I 


FIG.  34. 


center.  Draw  the  perpendicular  F  I,  and  the  line  F  C,  cutting  the  circle 
at  A ;  and,  with  the  radius  F  A  or  F  I,  describe  the  arc  A  I,  which  will  be  the 
arc  required. 

PKOB.  XXI. — To  fill  up  the  angle  of  a  straight  line  and  a  circle,  with  a 
circular  arc  to  join  the  circle  at  a  given  point  (Fig.  35). 

In  the  given  circle  B  A  draw  the  radius  to  A,  and  extend  it.  At  A 
draw  a  tangent,  meeting  the  given  line  at  D.  Bisect  the  angle  A  D  E,  so 
formed,  with  a  line  cutting  the  radius,  as  extended  at  F ;  and,  on  the  center 
F,  with  radius  F  A,  describe  the  arc  A  E,  which  will  be  the  arc  required. 


PKOB.  XXII. — To  describe  a  circular  arc  joining  two  circles,  and  to  touch 
one  of  them  at  a  given  point  (Fig.  36). 

Let  A  B  and  F  G  be  the  given  circles  to  be  joined  by  an  arc  touching  one 
of  them  at  F. 

Draw  the  radius  E  F,  and  produce  it  both  ways  ;  set  off  F  H  equal  to  the 
radius,  A  0,  of  the  other  circle ;  join  C  to  H,  and  bisect  it  with  the  perpen- 
dicular L  I,  cutting  E  F  at  I.  On  the  center  I,  with  radius  I  F,  describe  the 
arc  F  A,  which  will  be  the  arc  required. 


CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 


PROB.  XXIII. — To  find  the  arc  which  shall  be  tangent  to  a  given  point  on  a 
straight  line,  and  pass  through  a  given  point  outside  the  line  (Fig.  37). 

Erect  at  A,  the  given  point  on  the  given  line,  a  perpendicular  to  the  line. 
From  C,  the  given  point  outside  the  line,  draw  C  A,  and  bisect  it  with  a  per- 
pendicular. The  intersection  of  the  two  perpendiculars  at  a  will  be  the  center 
of  the  required  arc. 


a. 

FIG.  37. 


FIG.  38. 


PROB.  XXIV. — To  connect  two  parallel  lines  by  a  reversed  curve  composed 
of  two  arcs  of  equal  radii,  and  tangent  to  the  lines  at  given  points  (Fig.  38). 

Join  the  two  given  points  A  and  B,  and  divide  the  line  A  B  into  two  equal 
parts  at  C  ;  bisect  C  A  and  C  B  by  perpendiculars  ;  at  A  and  B  erect  perpen- 
diculars to  the  given  lines,  and  the  intersections  a  and  b  will  be  the  centers  of 
the  arcs  composing  the  required  curve. 

PROB.  XXV. — To  join  two  given  points 
in  two  given  parallel  lines  by  a  reversed 
curve  of  two  equal  arcs,  whose  centers  lie 
in  the  parallels  (Fig.  39). 

Join  the  two  given  points  A  and  B, 
and  divide  the  line  A  B  in  equal  parts  at 
C.  Bisect  A  C  and  B  C  by  perpendicu- 
lars ;  the  intersections  a  and  b  of  the 
parallel  lines,  by  these  perpendiculars, 
will  be  the  centers  of  the  required  arcs. 

PROB.  XXVI. — On  a  given  line,  to  construct  a  compound  curve  of  three 
arcs  of  circles,  the  radii  of  the  two  side  ones  being  equal  and  of  a  given  length, 


FIG. 


'/b 


H  / 

D 

FIG.  40. 


CONSTRUCTION   OF   GEOMETRICAL   PROBLEMS. 


15 


and  their  centers  in  the  given  line  ;  the  central  arc  to  pass  through  a  given 
point  on  the  perpendicular,  bisecting  the  given  line,  and  to  be  tangent  to  the 
other  two  arcs  (Fig.  40). 

Let  A  B  be  the  given  line,  and  C  the  given  point.  Draw  C  D  perpen- 
dicular to  A  B  ;  lay  off  A  a,  B  b,  and  C  c,  each  equal  to  the  given  radius  of  the 
side  arcs  ;  draw  a  c,  and  bisect  it  by  a  perpendicular ;  the  intersection  of  this 
line  with  the  perpendicular  C  D  will  be  the  required  center  of  the  central  arc 
e  C  er.  Through  a  and  b  draw  the  lines  D  e  and  D  e' ;  from  a  and  b,  with  the 
given  radius  equal  to  a  A  or  b  B,  describe  the  arcs  A  e  and  B  ef.  From  D,  as 
a  center,  with  a  radius  equal  to  C  D,  and,  consequently,  by  construction,  equal 
to  D  e  and  D  e',  describe  the  arc  e  C  e'.  The  entire  curve  A  e  C  e'  B  is  the 
compound  curve  required. 

t  It  will  be  observed  in  all  the  preceding  problems  that,  when  a  line  is  tangent 
to  a  curve,  the  center  of  that  curve  must  be  in  the  perpendicular  to  the  line 
at  its  tangent  point ;  and  that,  when  two  curves  are  tangent  to  each  other,  their 
centers  must  be  in  the  same  radial  line  passing  through  the  point  of  tangency. 

PROBLEMS    ON    POLYGONS   AND   CIRCLES. 

Three  lines  inclosing  a  spa^ce  form  a  triangle  (Fig.  41).  If  two  of  the  sides 
are  of  equal  length,  it  is  an  isosceles  triangle ;  if  all  three  are  of  equal  length, 


FIG.  41. 


FIG.  42. 


it  is  an  equilateral  triangle.  If  one  of  the  angles  is  a  right  angle,  it  is  a  right- 
angled  triangle,  and  if  no  two  of  the  sides  are  of  equal  length,  and  not  one  of 
the  angles  a  right  angle,  it  is  a  scalene 
triangle. 

PROB.  XXVII. — To  construct  an  isos- 
celes triangle  (Fig.  42). 


FIG.  43.  FIG.  44. 

Draw  any  line  as  a  base,  and,  from  each  extremity  as  a  center,  with  equal 
radius,  describe  intersecting  arcs.  Draw  a  line  from  each  extremity  of  the 
base  to  this  point  of  intersection,  and  the  figure  is  an  isosceles  triangle. 

PROB.  XXVIII. — To  construct  an  equilateral  triangle  (Fig.  43). 


16 


CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 


Draw  a  base  line,  and  from  each  extremity  as  a  center,  with  a  radius  equal 
to  the  base  line,  describe  intersecting  arcs.  Draw  lines  from  the  extremi- 
ties of  the  base  to  this  point  of  intersection,  and  the  figure  is  an  equilateral 
triangle. 

PKOB.  XXIX. — To  construct  a  right-angled  triangle  (Fig.  44). 

Construct  a  right  angle  by  any  one  of  the  methods  before  described. 
Draw  a  line  from  the  extremity  of  the  one  side  to  the  extremity  of  the  other 
side,  and  the  figure  is  a  right-angled  triangle. 

It  will  be  evident,  in  looking  at  any  right-angled  triangle,  that  the  side 
opposite  the  right  angle  is  longer  than  either  of  the  other  or  adjacent  sides ; 
this  side  is  called  the  hypothenuse. 

PROB.  XXX. — To  construct  a  triangle  equal  to  a  given  triangle. 

Let  ABC  (Fig.  45)  be  the  given  triangle. 

1st  Method  (Fig.  46). — Draw  a  base  line,  and  lay  off  its  length  equal  to 


FIG.  45. 


FIG.  46. 


A  B ;  from  one  of  its  extremities,  as  a  center,  with  a  radius  equal  to  A  C, 
describe  an  arc ;  and,  from  its  other  extremity,  with  a  radius  equal  to  B  C, 
describe  an  arc  intersecting  the  first.  Draw  lines  from  the  extremities  to  the 
point  of  intersection,  and  the  triangle  equal  to  A  B  C  is  complete. 

2d  Method  (Fig.  47). — Draw  a  base  line,  as  before,  equal  to  A  B.     At  one 

C 


FIG.  47. 


FIG.  48. 


extremity  construct  an  angle  equal  to  C  A  B,  and  at  the  other  an  angle  equal  to 
C  B  A.     The  sides  of  these  angles  will  intersect,  and  form  the  required  triangle. 

3d  Method  (Fig.  48).— Construct  an 
angle  of  the  triangle  equal  to  any  angle 
of  A  B  C,  say  the  angle  A  C  B.     On 
one  of  its  sides  measure  a  line  equal  to 
C  A,  and  on  the  other  side  one  equal  to 
C  B  ;  connect  the  two  extremiities  by  a 
line,  and  the  triangle  equal  to  A  B  C  is 
~~       complete. 
FIG.  49.  From  the  above  constructions  it  will 


CONSTRUCTION   OF  GEOMETRICA 


17 


FIG.  50. 


foe  seen  that,  if  the  three  sides  of  a  triangle,  or  two  sides  and  the  included  an- 
gle, or  one  side  and  the  two  adjacent  angles  are  known,  the  triangle  can  be 
constructed. 

Construct  a  triangle,  ABC  (Fig.  49).  Extend  the  base  to^E;"8m^Sraw 
B  D  parallel  to  A  C.  As  A  C  has  the  same  inclination  to  C  B  that  B  D  has, 
the  angle  C  B  D  is  equal  to  the  angle  A  C  B.  As  A  C  has  the  same  inclina- 
tion to  A  E  that  B  D  has,  the  angle  D  B  E  is  equal  to  C  A  B.  That  is, 
the  two  angles  formed  outside  the  triangle  are  equal  to  the  two  inside  at  A  and 
C  ;  and  the  three  angles  at  B  are  equal  to  the  three  angles  of  the  triangle,  and 
their  sum  is  equal  to  two  right  angles.  There- 
fore, the  sum  of  the  three  angles  of  a  trian- 
gle is  equal  to  two  right  angles. 

On  one  side  of  a  triangle  (Fig.  50)  con- 
struct a  triangle  equal  to  the  first,  with  op- 
posite sides  parallel. 

The    exterior    lines  of    the    two  triangles 

form  a  four-sided  or  quadrilateral  figure,  of  which  the  opposite  sides  are  equal 
and  parallel,  and  the  opposite  angles  equal.  This  figure  is  called  a  parallelo- 
gram, and  the  line  C  B,  extending  between  opposite  angles,  is  a  diagonal. 

On  the  hypothenuse  of  a  right-angled  triangle  (Fig.  51)  construct  another 
equal  to  it,  and  the  exterior  lines  form  a  parallelogram,  which,  as  all  the  angles 
are  right  angles,  is  called  a  rectangle.  If 
the  four  sides  are  all  equal,  it  is  called  a 
square. 

A  parallelogram  in  which  all  the  sides 
are  equal,  but  the  angles  not  right  angles, 
is  called  a  rhombus  (Fig.  52) ;  if  only  the 
opposite  sides  are  equal,  it  is  called  a  rhom- 
boid ;  if  only  two  sides  are  parallel,  the 
figure  is  a  trapezium  (Fig.  53). 

Describe  a  circle  (Fig.  54).  Draw  a  diameter,  and  erect  on  its  center  C  the 
perpendicular  C  F.  Draw  at  any  angle  with  the  diameter  the  line  C  A.  Draw 
D  H  and  A  B  perpendicular  to  the  diameter,  the  first  from  the  intersection  of 
the  line  C  A  with  the  circumference,  the  other  from  the  extremity  B  of  the 


FIG.  51. 


FIG.  52. 


FIG.  53. 


diameter.  Draw  D  G  and  E  F  perpendicular  to  the  radius  C  F,  one  from  the 
point  D,  the  other  from  the  extremity  of  the  radius  C  F.  The  angles  DOG 
and  D  C  H  are  complements  of  each  other  ;  that  is,  together  they  form  a 
right  angle,  as  it  completes  with  it  a  right  angle.  The  line  D  H  is  the 

2 


18 


CONSTRUCTION  OF   GEOMETRICAL  PROBLEMS. 


sine  of  the  angle  D  C  H  and  the  cosine  of  D  C  G.  D  G  is  the  sine  of  the 
angle  D  C  G  and  the  cosine  of  D  C  H.  A  B  is  the  tangent  of  the  angle 
.DOB  and  the  cotangent  of  D  C  G.  E  F  is  the  tangent  of  the  angle  DOG 
and  the  cotangent  of  D  C  H.  A  C  is  the  secant  of  the  angle  D  C  H  and 
the  cosecant  of  D  C  G.  C  E  is  the  secant  of  the  angle  DOG  and  the  cosecant 
of  D  C  H.  H  B  is  the  versed  sine  of  the  angle  D  C  H,  and  G  F  of  D  C  G. 

It  will  be  observed  that  the  angles  of  the  triangle  D  0  H  are  equal  to  those 
of  A  C  B,  and  that,  if  we  suppose  C  A  to  be  the  radius  of  a  larger  circle,  the 
arcs,  and  consequently  the  half -cords  or  sines  D  H  and  A  B,  will  be  propor- 
tionate to  the  radii ;  that  is,  D  H  will 
A  be  to  A  B  as  C  D  is  to  C  A. 

Triangles  which  have  equal  angles 
have  their  sides  proportional,  and  are 
called  similar.  This  is  demonstrable  of 
other  triangles  than  the  right-angled 
ones  in  the  figure. 

Take  any  figure  (Fig.  55)  of  more 
than  three  sides  bounded  by  straight 


FIG.  54. 


FIG.  55. 


lines,  and  from  any  angle  draw  lines  to  the  opposite  angles.  The  figure  will 
be  divided  into  as  many  triangles  as  there  are  sides,  less  two,  and  the  sum  of 
the  angles  of  the  figure  will  be  equal  to  as  many  times  two  right  angles  as 
there  are  sides,  less  two. 

If  another  figure  were  made  with  similar  triangles,  similarly  placed,  the 
two  figures  would  be  similar. 

Polygons,  or  many-sided  figures,  are  similar  when  their  angles  are  equal  to 
each  other  and  similarly  placed,  and  their  homologous  sides,  or  sides  including 
these  angles,  proportional. 


FIG.  56. 


FIG.  57. 


FIG.  58. 


FIG.  59. 


On  this  principle  of  similarity  of  figures  the  science  of  drawing  is  based. 
With  a  scale  of  equal  parts,  one  inch  on  paper,  for  instance,  representing  a 


CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 


19 


foot,  a  yard,  or  a  mile,  in  nature,  the  figure  drawn  on  that  scale  will  represent 
the  object  accurately  in  reduced  form  ;  and  measurements  may  be  made  in  de- 
tail by  the  scale  as  well  as  from  the  natural  object  in  the  shop  or  on  the  estate. 

Polygons,  with  their  sides  and  angles  equal,  are  called  regular  polygons 
(Figs.  56,  57,  58,  59). 

Regular  polygons  are  easily  constructed  by  means  of  circles,  whose  circum- 
ferences are  divided  into  the  number  of  sides  required,  with  chords  drawn 
representing  the  sides.  As  the  circle  is  then 
outside  the  polygon,  the  circle  is  said  to  be 
described  about  it,  while  the  polygon  is  in- 
scribed within  the  circle.  If  the  polygon  is 
described  about  the  circle,  its  sides  are  tan- 
gent to  it. 

PROB.  XXXI. — To  describe  a  circle  about  a 
triangle  (Fig.  60). 

Bisect  two  of  the  sides  A  B,  A  0,  of  the  tri- 
angle at  E,  F ;  from  these  points  draw  perpen- 
diculars cutting  at  K.  From  the  center  K, 
with  K  A  as  radius,  describe  the  circle  ABC, 
as  required. 

PROB.  XXXII. — To  inscribe  a  circle  in  a  triangle  (Fig.  61). 

Bisect  two  of  the  angles  A,  0,  of  the  triangle  A  B  C,  by  lines  cutting  at 
D  ;  from  D  draw  a  perpendicular  D  E  to  any  side,  as  A  0 ;  and  with  D  E  as 
radius,  from  the  center  D,  describe  the  circle  required. 

When  the  triangle  is  equilateral,  the  center  of  the  circle  is  more  easily  found 
by  bisecting  two  of  the  sides,  and  drawing  perpendiculars,  as  in  the  previous 
problem.  Or,  draw  a  perpendicular  from  one  of  the  angles  to  the  opposite 
side,  and  from  the  side  set  off  one  third  of  the  length  of  the  perpendicular. 


FIG.  60. 


FIG.  62. 

PROB.  XXXIII. — To  inscribe  a  square  in  a  circle ;  and  to  describe  a  circle 
about  a  square  (Fig.  62). 

To  inscribe  the  square.  Draw  two  diameters,  A  B,  0  D,  at  right  angles, 
and  join  the  points  A,  B,  0,  D,  to  form  the  square  as  required. 

To  describe  the  circle.  Draw  the  diagonals  A  B,  C  D,  of  the  given  square, 
cutting  at  E ;  on  E  as  a  center,  with  E  A  as  radius,  describe  the  circle  as 
required. 

In  the  same  way,  a  circle  may  be  described  about  a  rectangle. 


20 


CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 


PROB.  XXXIV. — To  inscribe  a  circle  in  a  square  ;  and  to  describe  a  square 
about  a  circle  (Fig.  63). 

To  inscribe  the  circle.  Draw  the  diagonals  A  B,  C  D,  of  the  giver  square, 
cutting  at  E ;  draw  the  perpendicular  E  F  to  one  of  the  sides,  and  with  the 
radius  E  F,  on  the  center  E,  describe  the  circle. 

To  describe  the  square.  Draw  two  diameters  A  B,  C  D,  at  right  angles, 
and  produce  them  ;  bisect  the  angle  D  E  B  at  the  center  by  the  diameter  F  G, 
and  through  F  and  G  draw  perpendiculars  A  C,  B  D,  and  join  the  points  A,  D, 
and  B,  C,  where  they  cut  the  diagonals,  to  complete  the  square. 

PROB.  XXXV. — To  inscribe  a  pentagon  in  a  circle  (Fig.  64). 

Draw  two  diameters,  A  C,  B  D,  at  right  angles ;  bisect  A  0  at  E,  and 
from  E,  with  radius  E  B,  cut  A  C  at  F  ;  from  B,  with  radius  B  F,  cut  the 


F 

FIG.  63. 


B 


FIG.  64. 


FIG.  65. 


circumference  at  G  and  H,  and  with  the  same  radius  step  round  the  circle  to  I 
and  K  ;  join  the  points  so  found  to  form  the  pentagon. 

PROB.  XXXVI. — To  construct  a  regular  hexagon  upon  a  given  straight 
line  (Fig.  65). 

From  A  and  B,  with  a  radius  equal  to  the  given  line,  describe  arcs  cutting 
at  g;  from  g,  with  the  radius  g  A,  describe  a  circle  ;  with  the  same  radius  set 
off  from  A  the  arcs  A  G,  G  F,  and  from  B  the  arcs  B  D,  D  E.  Join  the 
points  so  found  to  form  the  hexagon. 

PROB.  XXXVII. — To  inscribe  a  regular  hexagon  in  a  circle  (Fig.  66). 


J> 


FIG.  66. 


FIG.  67. 


Draw  a  diameter,  A  B ;  from  A  and  B  as  centers,  with  the  radius  of  the 
circle  A  C,  cut  the  circumference  at  D,  E,  F,  G ;  draw  straight  lines  A  D, 
D  E,  etc.,  to  form  the  hexagon. 


CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 


21 


The  points  of  contact,  D,  E,  etc.,  may  also  be  found  by  setting  off  the 
radius  six  times  upon  the  circumference. 

PROB.  XXXVIII. — To  describe  a  regular  hexagon  about  a  circle  (Fig.  67). 

Draw  a  diameter,  A  B,  of  the  given  circle.  With  the  radius  A  D  from  A 
as  a  center,  cut  the  circumference  at  C  ;  join  A  C,  and  bisect  it  with  the 
radius  D  E ;  through  E  draw  F  G  parallel  to  A  C,  cutting  the  diameter  at  F, 
and  with  the  radius  D  F  describe  the  circle  F  H.  Within  this  circle  describe 
a  regular  hexagon  by  the  preceding  problem ;  the  figure  will  touch  the  given 
circle  as  required. 

PROB.  XXXIX. — To  construct  a  regular  octagon  upon  a  given  straight  line 
(Fig.  68). 

Produce  the  given  line  A  B  both  ways,  and  draw  perpendiculars  A  E,  B  F ; 
bisect  the  external  angles  at  A  and  B  by  the  lines  A  H,  B  C,  which  make 


A  B 

FIG.  68. 


FIG.  69. 


equal  to  A  B ;  draw  C  D  and  H  G  parallel  to  A  E  and  equal  to  A  B ;  and 
from  the  centers  G,  D,  with  the  radius  A  B,  cut  the  perpendiculars  at  E,  F, 
and  draw  E  F  to  complete  the  octagon. 

PROB.  XL. — To  convert  a  square  into  a  regular  octagon  (Fig.  69). 

Draw  the  diagonals  of  the  square  intersecting  at  e;  from  the  corners  A,  B, 
C,  D,  with  A  e  as  radius,  describe  arcs  cutting  the  sides  at  g  h,  etc. ;  join  the 
points  so  found  to  complete  the  octagon. 

PROB.  XLI. — To  inscribe  a  regular  octagon  in  a  circle  (Fig.  70). 


FIG.  70. 


FIG.  71. 


Draw  two  diameters,  A  C,  B  D,  at  right  angles  ;  bisect  the  arcs  A  B, 
B  C,  etc.,  at  e,  /,  etc.;  and  join  A  e,  e  B,  etc.,  for  the  inscribed  figure. 


22 


CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 


PKOB.  XLII. — To  describe  a  regular  octagon  about  a  circle  (Fig.  71). 

Describe  a  square  about  the  given  circle  A  B ;  draw  perpendiculars  h  k, 
etc.,  to  the  diagonals,  touching  the  circle.  The  octagon  so  formed  is  the 
figure  required. 

Or,  to  find  the  points  h,  k,  etc.,  cut  the  sides  from  the  corners  of  the 
square,  as  in  Prob.  XL. 

PKOB.  XLIII. — To  inscribe  a  circle  within  a  regular  polygon. 

When  the  polygon  has  an  even  number  of  sides,  as  in  Fig.  72,  bisect  two 
opposite  sides  at  A  and  B,  draw  A  B,  and  bisect  it  at  C  by  a  diagonal  D  E 
drawn  between  opposite  angles  ;  with  the  radius  C  A  describe  the  circle  as 
required. 

When  the  number  of  sides  is  odd,  as  in  Fig.  73,  bisect  two  of  the  sides 
at  A  and  B,  and  draw  lines  A  E,  B  D,  to  the  opposite  angles,  intersecting 
at  C ;  from  C,  with  C  A  as  radius,  describe  the  circle  as  required. 


FIG.  72. 


FIG.  73. 


PROB.  XLIV. — To  describe  a  circle  without  a  regular  polygon. 

When  the  number  of  sides  is  even,  draw  two  diagonals  from  opposite 
angles,  like  D  E  (Fig.  72),  to  intersect  at  C ;  and  from  C,  with  C  D  as  radius, 
describe  the  circle  required. 

When  the  number  of  sides  is  odd,  find  the  center  C  (Fig.  73)  as  in  last 
problem,  and,  with  C  D  as  radius,  describe  the  circle. 

The  foregoing  selection  of  problems  on  regular  figures  is  the  most  useful 
in  mechanical  practice  on  that  subject.  Several  other  regular  figures  may  be 
constructed  from  them  by  bisection  of  the  arcs  of  the  circumscribing  circles. 
In  this  way  a  decagon,  or  ten-sided  polygon,  may  be  formed  from  the  penta- 
gon by  the  bisection  of  the  arcs  in  Prob.  XXXV,  Fig.  64.  Inversely,  an 
equilateral  triangle  may  be  inscribed  by  joining  the  alternate  points  of  division 
found  for  a  hexagon. 

The  constructions  for  inscribing  regular  polygons  in  circles  are  suitable 
also  for  dividing  the  circumference  of  a  circle  into  a  number  of  equal  parts. 
To  supply  a  means  of  dividing  the  circumference  into  any  number  of  parts, 
including  cases  not  provided  for  in  the  foregoing  problems,  the  annexed  table 
of  angles  relating  to  polygons,  expressed  in  degrees,  will  be  found  of  general 
utility.  In  this  table,  the  angle  at  the  center  is  found  by  dividing  360°,  the 
number  of  degrees  in  a  circle,  by  the  number  of  sides  in  the  polygon,  and  by 
setting  off  round  the  center  of  the  circle  a  succession  of  angles  by  means  of 


CONSTRUCTION   OF  GEOMETRICAL  PROBLEMS. 


23 


the  protractor,  equal  to  the  angle  in  the  table  due  to  a  given  number  of  sides  : 
the  radii  so  drawn  will  divide  the  circumference  into  the  same  number  of 
parts.  The  triangles  thus  formed  are  termed  the  elementary  triangles  of  the 
polygon. 

TABLE   OE   POLYGONAL   ANGLES. 


Number  of  Sides  of  Kegu- 

lar  Polygon  ;  or  number 

Angle  at 

Number  of  Sides  of 

Angle  at 

of  equal  parts  of  the  cir- 

Center. 

Kegular  Polygon. 

Center. 

cumference. 

No. 

Degrees. 

No. 

Degrees. 

3 

120 

12 

30 

4 

90 

13 

27* 

5 

72 

14 

25f 

6 

60 

15 

24 

7 

51-f 

16 

22J 

8 

45 

17 

21* 

9 

40 

18 

20 

10 

36 

19 

18|f 

11 

32* 

20 

18 

CONSTRUCTION    OF    THE    ELLIPSE,     PAEABOLA,     HYPERBOLA,    CYCLOID,    EPICY- 
CLOID,   INVOLUTE,    AND   SPIRAL. 

An  ellipse  is  an  oval-shaped  curve  (Fig.  74),  in  which,  if  from  any  point, 
P,  lines  be  drawn  to  two  fixed  points,  F  and  F',  foci,  their  sum  will  always  be  the 
same.  The  line  A  B  passing  through 
the  foci  is  the  transverse  axis,  and 
the  perpendicular  C  D  at  the  cen- 
ter of  it  is  the  conjugate  axis. 

PROB.  XLV.  —  To  construct 
an  ellipse,  the  axes  being  known 
(Fig.  75). 

1st  Method. — Let  the  two  axes 
be  the  lines  A  B  and  C  D.  From 
0  as  a  center,  with  a  radius  equal 
to  E  B  (half  the  transverse  axis), 
describe  an  arc  cutting  this  axis 
at  two  points,  F  and  F',  which  are 
the  foci.  Insert  a  pin  in  each  of  the  foci,  and  loop  a  thread  upon  them,  so 
that,  when  stretched  by  a  pencil  inside  the  loop,  the  point  of  the  pencil 
will  coincide  with  C.  Move  the  pencil  round,  keeping  the  loop  evenly 
stretched,  and  it  will  describe  an  ellipse.  This  construction  follows  the 
definition  above  given  of  an  ellipse,  that  the  sum  of  the  distances  of  every 
point  of  the  curve  from  the  foci  is  equal.  It  is  seldom  used  by  the  draughts- 
man, as  it  is  difficult  to  keep  a  thread  evenly  stretched ;  but  for  gardeners, 
laying  out  beds  or  plots,  it  is  very  convenient  and  sufficiently  accurate. 


CONSTRUCTION  OF  GEOMETRICAL  PROBLEMS. 


2d  Method.— Carpenters,  almost  invariably,  lay  out  an  ellipse  by  means 
of  a  trammel  (Fig.  76),  which  consists  of  a  rectangular  cross,  E  Gr  F  H, 

with  guiding  grooves,  in  which 
metal  rods,  attached  to  slides 
on  a  bar,  are  fitted  so  as  to 
move  easily  and  uniformly.  In 
describing  the  ellipse,  the  tram- 
mel is  placed  with  its  grooves 
on  the  lines  of  the  axes.  Ad- 
just the  metal  points,  Tc  and  Z, 
which  slide  in  the  grooves,  so 
as  to  have  between  them  a  dis- 
tance equal  to  half  the  conju- 
gate axis,  and  make  the  dis- 
tance from  k  to  m  (the  position 

on  the  bar  of  the  pencil  or  marker)  equal  to  half  the  transverse  axis.     Kevolve 
the  bar,  keeping  the  points  Tc  and  I  always  in  the  grooves,  and  the  pencil  will 

describe  an  ellipse.  Xeat 
instruments  of  this  sort  are 
made  for  the  use  of  the 


draughtsman,  but,  for  of- 
fices where  curves  of  this 
sort  are  required  but  little, 
a  substitute  for  the  tram- 
mel can  be  had  in  a  strip  of 
paper  (Fig.  77),  by  marking 
the  straight  edge  at  a  and  b 
and  c,  the  distance  c  a  being 
made  equal  to  half  the  trans- 
verse axis,  and  the  distance 
c  b  to  half  the  conjugate 


FIG.  77. 


CONSTRUCTION  OF  GEOMET 


25 


axis.  Revolving  the  strip  of  paper,  keeping  b  on  the  line  of  the  transverse 
axis,  and  c  on  the  line  of  the  conjugate  axis,  and  dotting  the  positions  of  a  at 
short  intervals,  enough  points  of  the  curve  will  be  determined  through  which 
the  ellipse  may  be  drawn  readily. 

PEOB.  XLVI.  —  To   describe  an  ellipse  approximately,  by  means  of  cir- 
cular arcs. 

First,  with  arcs  of  two 
radii  (Fig.  78).  Take  the 
difference  of  the  transverse 
and  conjugate  axes,  and  set 
it  off  from  the  center  0  to  / 
a  and  c,  on  0  A  and  0  C  ; 
draw  a  c,  and  set  off  half  a  c 
to  d;  draw  d  i  parallel  to 
a  c,  set  off  0  e  equal  to  0  d, 
join  e  i,  and  draw  e  m,  d  m, 
parallels  to  d  i,  i  e.  On  cen- 
ter m,  with  radius  m  C,  de- 
scribe an  arc  through  C,  and 
from  center  i  describe  an  arc 
through  D  ;  on  centers  dy  e, 
also,  describe  arcs  through  A  and  B.  The  four  arcs  thus  described  form 
approximately  an  ellipse.  This  method  does  not  apply  satisfactorily  when  the 
conjugate  axis  is  less  than  two  thirds  of  the  transverse  axis. 

Second,  with  arcs  of  three  radii  (Fig.  79).     On  the  transverse  axis  A  B, 
draw  the  rectangle  B  G,  equal  in  height  to  0  C,  half  the  conjugate  axis. 


Extend  0  C  above  and  below  the  rectangle.  Draw  Gr  D  perpendicular  to  A  0, 
intersecting  0  C  extended  at  D.  Set  off  0  K  equal  to  0  C,  and  on  A  K  as  a 
diameter  describe  the  semicircle  A  N  K ;  draw  a  radius  parallel  to  0  0, 


26  CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 

intersecting  the  semicircle  at  N  and  the  line  G  E  at  P ;  set  off  0  M  equal  to 
P  N,  and  on  D  as  a  center,  with  a  radius  D  M,  describe  an  arc ;  from  A  and 
B  as  centers,  with  a  radius  0  L,  intersect  this  arc  at  a  and  b.  The  points 
H,  #,  D,  b,  H',  are  the  centers  of  the  arcs  required ;  produce  the  lines  a 
H,  D  a.  D  b,  b  H',  and  the  spaces  inclosed  determine  the  lengths  of  each  arc. 

This  process  works  well  for  nearly  all  proportions  of  ellipses.  It  is  em- 
ployed in  striking  out  vaults  and  stone  bridges. 

PROB.  XLVII. — To  draw  a  tangent  to  an  ellipse  through  a  given  point  in 
the  curve  (Fig.  80). 

From  the  given  point  T  draw  straight  lines  to  the  foci  F,  F';  produce  F  T 


beyond  the  curve  to  c,  and  bisect  the  exterior  angle  c  T  F'  by  the  line  T  d. 
This  line  T  d  is  the  tangent  required. 

PROB.  XLVIII. — To  draw  a  tangent  to  an  ellipse  from  a  given  point  with- 
out the  curve  (Fig.  81). 

From  the  given  point  T  as  center,  with  a  radius  equal  to  its  distance  from 
the  nearest  focus  F,  describe  an  arc ;  from  the  other  focus  F',  with  the  trans- 


;**  \K 


verse  axis  as  radius,  cut  the  arc  at  K,  L,  and  draw  K  F',  L  F',  cutting  the 
curve  at  M,  N ;  then  the  lines  T  M,  T  N,  are  tangents  to  the  curve. 

The  Parabola. 

The  parabola  may  be  defined  as  an  ellipse  whose  transverse  axis  is  infinite  5 
its  characteristic  is  that  every  point  in  the  curve  is  equally  distant  from  the 
directrix  E  N,  and  the  focus  F  (Fig.  82). 

PROB.  XLIX. — To  construct  a  parabola  when  the  focus  and  directrix  are 
given. 


CONSTRUCTION  OF  GEOMETRICAL  PROBLEMS. 


27 


1st  Method  (Fig.  82). — Through  the  focus  F  draw  the  axis  A  B  perpendicu- 
lar to  the  directrix  E  N,  and  bisect  A  F  at  e,  then  e  is  the  vertex  of  the  curve. 
Through  a  series  of  points,  C,  D,  E,  on  the  directrix,  draw  parallels  to  A  B ; 
connect  these  points,  C,  D,  E,  with  the  focus  F,  and  bisect  by  perpendiculars 
the  lines  F  C,  F  D,  F  E.  The  intersections  of  these  perpendiculars  with  the  par- 
allels will  give  points,  C',  D',  E',  in  the  curve,  through  which  trace  the  parabola. 

2d  Method  (Fig.  83). — Place  a  straight-edge  to  the  directrix  E  N,  and 
apply  to  it  a  square  LEG;  fasten  at  G  one  end  of  a  cord,  equal  in  length 


FIG.  82. 


FIG.  83. 


ri 


to  E  G ;  fix  the  other  end  to  the  focus  F ;  slide  the  square  steadily  along 
the  straight-edge,  holding  the  cord  taut  against  the  edge  of  the  square  by  a 
pencil,  D,  and  it  will  describe  the  curve. 

PKOB.  L. — To  construct  a  parabola  when  the  vertex,  the  axis,  and  a  point 
of  the  curve  are  given  (Fig.  84). 

Let  A  be  the  vertex,  A  B  be 
the  axis,  and  M  the  given  point 
of  the  curve. 

Construct  the  rectangle  A  B- 
M  0.  Divide  M  0  into  any  num- 
ber of  equal  parts,  four,  for  in- 
stance ;  divide  A  C  in  like  man- 
ner ;  draw  A  1,  A  2,  A  3  ;  through 
1',  2',  and  3',  draw  lines  parallel 
to  the  axis.  The  intersections  I,  II,  and  III,  of  these  lines  are  points  in  the 
required  curve. 

The  Hyperbola. 

An  hyperbola  is  a  curve  from  any  point  P,  in  which,  if  two  straight  lines 
be  drawn  to  two  fixed  points,  F,  F',  the  foci,  their  difference  shall  always  be 
the  same. 


CONSTRUCTION  OF  GEOMETRICAL  PROBLEMS. 


PKOB.  LI.  — To  describe  an  hyperbola  (Fig.  85). 

From  one  of  the  foci  F,  with  an  assumed  radius,  describe  an  arc,  and  from 
the  other  focus  F',  with  another  radius  exceeding  the  former  by  the  given 
difference,  describe  two  small  arcs,  cutting  the  first  as  at  P  and  p.  Let  this 
operation  be  repeated  with  two  new  radii,  taking  care  that  the  second  shall 
exceed  the  first  by  the  same  difference  as  before,  and  two  new  points  will  be 
determined ;  and  this  determination  of  points  in  the  curve  may  thus  be  con- 
tinued till  its  track  is  obvious.  By  making  use  of  the  same  radii,  but  trans- 
posing, that  is,  describing  with  the  greater  about  F,  and  the  less  about  F',  we 
have  another  series  of  points  equally  belonging  to  the  hyperbola,  and  answer- 
ing the  definition  ;  so  that  the  hyperbola  consists  of  two  separate  branches. 


FIG.  85. 


FIG.  86. 


The  curve  may  be  described  mechanically  (Fig.  86). — By  fixing  a  ruler 
to  one  focus  F',  so  that  it  may  be  turned  round  on  this  point,  connect  the 
other  extremity  of  the  ruler  R  to  the  other  focus  F  by  a  cord  shorter  than  the 
whole  length  F7  R  of  the  ruler  by  the  given  difference  ;  then  a  pencil  P  keep- 
ing this  cord  always  stretched,  and  at 
the  same  time  pressing  against  the 
edge  of  the  ruler,  will,  as  the  ruler 
revolves  around  F',  describe  an  hy- 
perbola, of  which  F  F'  are  the  foci, 
and  the  differences  of  distances  from 
these  points  to  every  point  in  the 
curve  will  be  the  same. 

PEOB.  LII. — To  draw  a  tangent  to 
any  point  of  an  hyperbola  (Fig.  87). 

Let  P  be  the  point.     On  F'  P  lay 
off  P  p,  equal  to  F  P ;  draw  the  line 
F  p  ;  from  P  let  fall  a  perpendicular 
Fio  87  on  this  line,  P  p,  and  it  will  be  the 

tangent  required. 

The  three  curves,  the  ellipse,  the  parabola,  and  the  hyperbola,  are  called 
conic  sections,  as  they  are  formed  by  the  intersections  of  a  plane  with  the  sur- 
face of  a  cone.  See  CONSTRUCTION  OF  THE  CONIC  SECTIONS. 


CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 


If  the  cone  be  cut  through  both  its  sides  by  a  plane  not  parallel  to  the 
base,  the  section  is  an  ellipse ;  if  the  intersecting  plane  be  parallel  to  the  side 
of  the  cone,  the  section  is  a  parabola ;  if  the  plane  have  such  a  position  that, 
when  produced,  it  meets  the  opposite  cone,  the  section  is  an  hyperbola.  The 
opposite  cone  is  a  reversed  cone  formed  on  the  apex  of  the  other  by  the  con- 
tinuation of  its  sides. 

The  Cycloid. 

The  cycloid  is  the  curve  described  by  a  point  in  the  circumference  of  a 
circle  rolling  on  a  straight  line. 

PKOB.  LIII. — To  describe  a  cycloid  (Fig.  88). 

Draw  the  straight  line  A  B  as  the  base  ;  describe  the  generating  circle  tan- 
gent at  the  center  of  this  line,  and  through  the  center  0  draw  the  line  E  E 
parallel  to  the  base ;  let  fall  a  perpendicular  from  C  upon  the  base ;  divide 


the  semi-circumference  into  any  number  of  equal  parts,  for  instance,  six ;  lay 
off  on  A  B  and  0  E  distances  C  I/  V  2'. . .,  equal  to  the  divisions  of  the 
circumference  ;  draw  the  chords 
D  1,  D  2.  .  .  ;  from  the  points 
1',  2',  3'.  .  .on  the  line  C  E, 
with  radii  equal  to  the  generat- 
ing circle,  describe  arcs ;  from 
the  points  1',  2',  3',  4',  5',  on 
the  line  B  A,  and  with  radii 
equal  successively  to  the  chords 
D  1,  D  2,  D  3,  D  4,  D  5,  describe 
arcs  cutting  the  preceding,  and 
the  intersections  will  be  points 
of  the  curve  required. 

2d  Method  (Fig.  89).— Let 
0  9'  be  the  base-line,  0  4  9  the 
half  of  the  generating  circle ; 
divide  the  half  circle  into  any 
number  of  equal  parts,  say  nine, 
and  draw  the  chord  0  1,  0  2,  Fl0'  89* 

0  3,  etc.  ;  lay  off  on  the  base  0  1',  I'  2',  2'  3' ,  equal  respectively  to  the 

length  of  one  of  the  divisions  of  the  half  circle  0  1 ;  draw  through  the  points 


30 


CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 


1',  2',  3' lines  parallel  to  the  chords  0  1,0  2,  0  3 ;  the  intersections 

I,  II,  III of  these  lines  are  centers  of  the  arcs  0  a,  al),  I  c ,  of  which 

the  cycloid  is  composed. 

The  Epicycloid. 

The  epicycloid  is  formed  by  a  point  in  the  circumference  of  a  circle  revolv- 
ing either  externally  or  internally  on  the  circumference  of  another  circle  as 


PKOB.  L1V. — To  describe  an  epicycloid. 

Let  us  in  the  first  place  take  the  exterior  curve.  Divide  the  circumfer- 
ence A  B  D  (Fig.  90)  into  a  series  of  equal  parts  1,  2,  3 ,  beginning  from 

the  point  A ;  set  off  in  the  same  manner,  upon  the  circle  A  M,  A  N,  the  divis- 
ions 1',  2',  3' equal  to  the  divisions  of  the  circumference  A  B  D.  Then, 

as  the  circle  A  B  D  rolls  upon  the  circle  A  M  A  N,  the  points  1,  2,  3  will 
coincide  successively  with  the  points  1',  2',  3';  and,  drawing  radii  from  the 


point  0  through  the  points  1',  2',  3',  and  also  describing  arcs  of  circles  from 

the  center  0,  through  the  points  1,  2,  3 ,  they  will  intersect  each  other 

successively  at  the  points  c,  d,  e Take  now  the  distance  1  to  c,  and  set 

it  off  on  the  same  arc  from  the  point  of  intersection  i,  of  the  radius  A  C  ; 
in  like  manner,  set  off  the  distance  2  to  d,  from  b  to  A2,  and  the  distance  3  to 
e,  to  A8,  and  so  on.  Then  the  points  A1,  A2,  A3,  will  be  so  many  points  in  the 
epicycloid  ;  and  their  frequency  may  be  increased  at  pleasure  by  shortening 


^»^\. 

CONSTRUCTION   OF   GEOMETRICAL   PROBLEMS.  31 

the  divisions  of  the  circular  arcs.  Thus  the  form  of  the  curve  may  be  deter- 
mined to  any  amount  of  accuracy,  and  completed  by  tracing  a  line  through 
the  points  found. 

As  the  distances  1  to  c, which  are  near  the  commencement  of  the 

curve,  must  be  very  short,  it  may,  in  some  instances,  be  more  convenient  to 
set  off  the  whole  distance  i  to  1  from  c,  and  in  the  same  way  the  distance  b 
to  2  from  d  to  A2,  and  so  on.  In  this  manner  the  form  of  the  curve  is  the 
more  likely  to  be  accurately  defined. 

2d  Method. — To  find  the  points  in  the  curve,  find  the  positions  of  the 
center  of  the  rolling  circle  corresponding  to  the  points  of  contact  1',  2',  3', 
etc.,  which  may  be  readily  done  by  producing  the  radii  from  the  center  0, 

through  the  points  1',  2',  3', to  cut  the  circle  B  C.  From  these  centers 

describe  arcs  of  a  circle  with  the  radius  of  C  A,  cutting  the  corresponding 

arcs  described  from  the  center  0,  and  passing  through  the  points  1,2,  3, 

as  before.  The  intersections  of  these  arcs  at  A1,  A2,  A3, . . .  .give  points  of  the 
curve. 

When  the  moving  circle  A  B  D  is  made  to  roll  on  the  interior  of  the  cir- 
cumference A  M,  A  N,  as  shown  (Fig.  91),  the  curve  described  by  the  point 


x 


\ 


FIG.  91. 

A  is  called  an  interior  epicycloid.  It  may  be  constructed  in  the  same  way  as 
in  the  preceding  case,  as  may  be  easily  understood,  the  same  figures  and  letters 
of  reference  being  used  in  both  figures. 

The  Involute. 

The  involute  is  a  curve  traced  by  the  extremity  of  a  flexible  line  unwind- 
ing from  the  circumference  of  a  circle. 


32 


CONSTRUCTION   OF  GEOMETRICAL  PROBLEMS. 


PROB.  LV. — To  describe  an  involute. 

Divide  the  circumference  of  the  given  circle  (Fig.  92)  into  any  number  of 
equal  parts,  as  0,  1,  2,  3,  4, ;  at  each  of  these  points  draw  tangents  to  the 


FIG.  92. 

given  circle  ;  on  the  first  of  these  lay  off  the  distance  11',  equal  to  the  arc 

0  1 ;  on  the  second  lay  off  2  2',  equal  to  twice  the  arc  0  1  or  the  arc  0  2  : 

establish  in  a  similar  way  the  points  3',  4',  5', as  far  as  may  be  requisite, 

which  are  points  in  the  curve  required. 

It  may  be  remarked  that,  in  all  the  problems  in  which  curves  have  been 

determined  by  the  position  of  points,  the  more   numerous  the  points   thus 

fixed,  the  more  accurately  can  the 
curve  be  drawn. 

The  involute  curve  may  be 
described  mechanically  in  several 
ways.  Thus,  let  A  (Fig.  93)  be 
the  center  of  a  wheel  for  which 
the  form  of  involute  teeth  is  to 
be  found.  Let  m  n  a  be  a  thread 
lapped  round  its  circumference, 
having  a  loop-hole  at  its  extrem- 
ity, a;  in  this  fix  a  pin,  with 
which  describe  the  curve  or  in- 
volute a  b h,  by  unwinding 

the  thread  gradually  from  the  circumference,  and  this  curve  will  be  the  proper 

form  for  the  teeth  of  a  wheel  of  the  given  diameter. 

The  Spiral 
The  spiral  is  the  involute  of  a  circle  produced  beyond  a  single  revolution. 


CONSTRUCTION   OF  GEOMETRICAL   PROBLEMS. 


33 


PROB.  LVL— To  describe  a  spiral  (Fig.  94,  and  Fig.  95  of  the  primary  on 
a  larger  scale). 

Divide  the  circumference  of  the  primary  into  any  number  of  equal  parts, 
say  not  less  than  eight.  To  these  points  of  division  o,  e,  f,  i,  etc.,  draw  tangents, 
and  from  these  points  draw  a  succession  of  circular  arcs ;  thus,  from  o  e  lay 


FIG.  94. 


FIG.  95. 


off  o  g,  equal  to  the  arc  a  o  reduced  to  a  straight  line,  and  connect  a  and  g 
by  a  curve  ;  from  e,  with  the  radius  e  g,  describe  the  arc  g  h  ;  from  /  the  next 
arc,  and  so  on.  Continue  the  use  of  the  centers  successively  and  repeatedly 
to  the  extent  of  the  revolutions  required.  Thus  the  point  a  in  the  figure  is 
used  as  a  center  for  three  arcs,  b  I,  c  m,  d  n. 


USE  or  TKIAKGLE  A:NT>  SQTJABE. 

Right-angled  triangles  constructed  of  wood,  hard  rubber,  or  metal,  are  very 
useful  in  connection  with  a  straight-edge,  or  ruler,  in  drawing  lines  parallel 
or  perpendicular  to  other  lines. 

To  draw  lines  parallel  to  each  other,  place  any  edge  of  the  triangle  in  close 
contact  with  the  edge  of  the  ruler.     Hold  the  ruler  (Fig.  96)  firmly  with  the 
3 


34  CONSTRUCTION  OF  GEOMETRICAL  PROBLEMS. 

thumb  and  little  finger  of  the  left  hand,  and  the  triangle  with  the  other  three 
fingers ;  with  a  pencil  or  pen  in  the  right  hand,  draw  a  line  along  one  of  the 
free  edges  of  the  triangle  ;  withdraw  the  pressure  of  the  three  fingers  upon  the 


FIG.  96. 


triangle,  and  slide  it  along  the  edge  of  the  ruler,  keeping  the  edges  in  close 
contact ;  a  line  drawn  along  the  same  edge  of  the  triangle,  as  before,  will  be 
parallel  to  the  first  line.  If  the  edge  of  the  hypothenuse  of  the  triangle  be 
placed  in  contact  with  the  ruler,  lines  drawn  along  one  edge  of  the  triangle 
will  be  at  right  angles  to  those  drawn  along  the  other. 


FIG.  97. 


PROB.  LVII. — Through  a  given  point  to  draw  a  line  parallel  to  a  given 
line  (Fig.  97). 

Place  one  of  the  shorter  edges  of  the  triangle  along  the  given  line  A  B,  and 


CONSTRUCTION  OF  GEOMETRICAL  PROBLEMS. 


35 


bring  the  ruler  against  the  hypothenuse  ;  slide  the  triangle  up  along  the  edge 
of  the  ruler  until  the  upper  edge  of  the  ruler  is  sufficiently  near  to  the  given 


2k 

'V 

4 

-\ 

E                   D 

£ 

r^ 

FIG.  98. 


FIG.  99. 


point  C  to  allow  a  line  to  be  drawn  through  it.     Draw  the  line,  and  it  will  be 
parallel  to  A  B. 

If  the  triangle  be  slid  farther  up  along  the  edge  of  the  ruler,  and  a  line  be 
drawn  through  C  along  the  other  edge  of  the  triangle  (Fig.  98),  this  line  will 
be  perpendicular  to  A  B.  If  the  triangle  be  slid  still  farther  up  along  the 
edge  of  the  ruler,  and  a  third  line  be  drawn  touching  A  B,  the  figure  con- 
structed will  be  a  rectangle  ;  and  if  E  D 
be  laid  off  on  A  B,  equal  to  C  E,  the  fig- 
ure inclosed  is  a  square  (Fig.  99). 

It  will  be  seen  that  the  triangle  and 
ruler  afford  a  much  readier  way  of  draw- 
ing parallel  lines,  and  lines  at  right  an- 
gles, than  the  compasses  and  ruler,  and 
may  be  used  in  solving  the  following 
problems  : 

The  area  of  a  figure  is  the  quantity 
of  space  inclosed  by  its  lines. 

Construct  a  right  angle  (Fig.  100).  Divide  the  base  and  the  perpendicular 
by  dividers  into  any  number  of  equal  spaces  ;  for  instance,  eight  on  the  one 
and  five  on  the  other.  Construct  a  rectangle  with  this  base  and  perpendicu- 
lar, and  through  the  points  of  division  lay  off  lines  parallel  to  the  base  and 
perpendicular.  The  rectangle  will  be  divided  into  forty  equal  squares,  and 
its  measure  in  squares  will  be  the  divisions  eight  in  the  base,  multiplied  by 
the  five  in  the  perpendicular.  If  the  division  were  inches,  then  the  area  of 


FIG.  100. 


FIG.  101. 


B 

FIG.  102. 


this  rectangle  would  be  forty  square  inches ;  if  feet,  then  forty  square  feet. 
If  there  were  but  five  divisions  in  the  base  and  five  in  the  perpendicular, 
the  surface  would  be  twenty-five  squares.  Therefore,  a  rectangle  has  for  its 
measure  the  base  multiplied  by  its  adjacent  side  or  height. 


36  CONSTRUCTION   OF  GEOMETRICAL  PROBLEMS. 

Draw  a  diagonal,  and  the  rectangle  is  divided  into  two  equal  triangles. 
Each  triangle  must  therefore  have  for  its  measure  the  base  multiplied  by  half 
the  perpendicular,  or,  as  is  usually  said,  by  half  the  altitude. 

Take  any  triangle  (Fig.  101),  and  from  its  apex  draw  a  line  perpendicular 
to  the  base.  The  triangle  is  divided  into  two  right-angled  triangles,  which 
must  have  for  their  measure  A  D  x  £  C  D,  and  D  B  x  ^  C  D,  and  the  sum  of 
the  two  must  be  A  B  x  £  C  D. 

If  the  perpendicular  from  the  apex  falls  outside  the  triangle  (Fig.  102), 
then  the  triangles  B  D  C  and  ADC  will  have  for  their  measure  B  D  x  J  C  D 
and  A  D  x  £  C  D ;  and  as  the  origi- 
nal triangle  A  B  C  is  the  difference 
of  these  two  triangles,  its  measure  must 
be  A  B  x  £  C  D.  Every  triangle  must 
have  for  its  measure  the  base  multi- 
plied by  half  the  altitude,  and  it  makes 
no  difference  which  side  is  taken  as  the 
base. 

Construct  the  right-angled  triangle 
A  C  B  (Fig.   103),   and  let  fall    the  FlG-  103- 

perpendicular  C  D.  As  will  be  seen  by  the  equality  of  the  angles  compos- 
ing the  triangles,  the  perpendicular  divides  the  original  triangle  into  two  right- 
angled  triangles,  similar  to  each  other  and  to  the  original  triangle.  Therefore 


FIG.  104. 

A  D  is  to  C  D  as  C  D  is  to  B  D,  or,  expressed  by  signs,  A  D  :  C  D  :  :  C  D  : 
B  D ;  therefore,  by  the  Rule  of  Three,  A  D  x  B  D  =  C  D2;  that  is,  C  D  is  a 
mean  proportional  between  A  D  and  B  D.  So  that  the  perpendicular  let  fall 


CONSTRUCTION   OF  GEOMETRICAL  PROBLEMS. 


from  the  vertex  of  a  right  angle  upon  the  hypothenuse  of  the  triangle,  is  a 
mean  proportional  between  the  two  parts  of  the  hypothenuse  into  which  it  is 
divided  by  the  perpendicular. 

In  comparing  the  two  triangles  with  the  original  triangle,  A  C  is  a  mean 
proportional  between  A  D  and  A  B,  and  B  C  is  a  mean  proportional  between 
B  D  and  A  B  ;  that  is,  A  C2=A  Dx A  B 

BC2=BDxAB 

adding  the  two,  A  Ca+B  C2  =  (A  D+B  D)xA  B 

and  as  A  D  +  B  D  =  A  B,  we  have  A  Ca+  B  C2= A  B2 ;  that  is,  the  square 
on  the  hypothenuse  is  equal  to  the  sum  of  the  squares  on  the  other  two 
sides. 

Construct  squares  on  the  three  sides  of  a  right-angled  triangle  (Fig.  104). 

b 


FIG.  105. 


FlCr.     106. 


PROB.  LVIII. — To  construct  a  square  equal  to  one  half  of  a  given  square 
(Fig.  105). 


FIG.  107. 


FIG.  108. 


Construct  the  given  square,  and  draw  diagonals  in  it.  The  square,  abed, 
constructed  on  one  half  of  one  of  these  diagonals  will  be  equal  to  one  half  the 
given  square. 


38 


CONSTRUCTION  OF  GEOMETRICAL  PROBLEMS. 


PKOB.  LIX.  —  To  construct  a  square  equal  to  double  a  given  square 
(Fig.  106). 

Construct  a  square  on  one  of  the  diagonals  in  the  given  square,  or  en- 
close the  square  with  parallels  to  the  diagonals  of  the  square. 

PEOB.  LX. — To  construct  a  square  equal  to  three  times  a  given  square 
(Fig.  107). 

Extend  the  base  of  the  given  square,  and  lay  off  on  it  the  length  of  its 
diagonal.  Draw  a  line  from  the  point  at  which  this  diagonal  ends  to  the  ex- 
treme angle  of  the  square,  and  upon  this  line  erect  a  square,  which  will  be  the 
square  required. 

For  a  square  four  times  the  size  of  a  given  square,  make  the  base  double 
that  of  the  given  square. 

PKOB.  LXI. — To  construct  a  square  equal  to  five  times  a  given  square 
(Fig.  108). 

Extend  the  base  of  the  given  square,  making  the  extension  to  d  equal  to 
the  base  of  the  given  square.  From  d  draw  a  line  to  a,  and  on  this  line  con- 
struct a  square,  abed,  which  will  be  the  square  required. 


FIG.  109. 


Assuming  the  side  of  the  given  square  in  Figs.  105,  106,  107,  and  108  to 
be  the  radius  (or  diameter)  (Fig.  109)  of  a  given  circle,  then  the  side  of  the 
square  to  be  constructed  half,  twice,  three,  four,  or  five  times  the  size  of  the 
given  square  will  be  the  radii  (or  diameters)  of  the  circles  half,  twice,  three,  four, 
or  five  times  the  size  of  the  given  circle. 


CONSTRUCTION   OF   GEOMETRICAL  PROBLEMS. 


39 


PKOB.  LXII.  —  To  determine  how  much  is  added  to  a  given  square  by 
extending  its  base  and  constructing  a  square  thereon  (Fig.  110). 

p 


c 


FIG.  110. 


H 


K 


J 


Let  a  represent  the  length  C  D  of  the  base  of  the  given  square.     Its  square 
will  be  a  X  a  or  a?  . 

Extend  the  base  C  D  by  a  certain  length,  D  G,  represented  by  I.     Then 
the  new  square  (a  +  b)  x  (a  +  b)  will  be  made  up  of  the  old  square,  or     a2 

and  two  rectangles,  D  G  E  H  and  P  E  K  L,  or  2  (a  x  b)  or  2  a  b 
and  one  square,  E  H  K  J,  or  b  x  b  or     b2 


PROB.  LXIII.  —  To  determine  how  much  is  taken  from  the  area  of  a  given 
square,  by  reducing  its  base  and  constructing  a  square  (Fig.  110). 

Let  a  represent  the  length  C  G  of  the  base  of  the  given  square.     Reduce 
C  G  by  a  certain  length,  G  D,  to  be  represented  by  b. 

Then  the  new  square  (a—  b)*  will  be  the  old  square,  or        a9 
diminished  by  two  rectangles,  D  G  J  K  and  P  L  J  H,  or—  2  a  b 
excepting  one  square,  E  H  J  K,  or  b  x  b  or     4-  b* 


The  last  two  constructions,  in  default  of  a  table  of  squares,  may  often  be 
found  of  use. 


DRAWING    INSTRUMENTS. 

THE  simple  drawing  instruments,  already  illustrated  and  applied  in  the 
construction  of  the  preceding  problems,  together  with  scales  of  equal  parts, 
a  protractor  and  a  drawing  pen,  are  all  the  instruments  essential  for  topo- 
graphical or  mechanical  drawing.  It  is  often  convenient,  for  facility  in  work- 
ing, to  have  compasses  of  varied  sizes  and  modifications,  and  these,  together 
with  an  assortment  of  rulers,  triangles,  squares,  scales,  and  protractors, 
adapted  to  varied  work,  are  included  in  boxes  of  drawing  instruments  as 
furnished  by  dealers.  The  smaller  rulers  and  triangles,  as  furnished,  are 
generally  of  hard  rubber,  and  the  larger  of  wood.  As  it  is  often  incon- 
venient to  carry  long  rulers,  and  difficult  to  procure  them  ready-made,  the 
draughtsman  may  have  to  depend  on  a  carpenter  or  joiner  for  them. 

Eulers  should  be  of  close-grained,  thoroughly  -  seasoned  wood,  such  as 
mahogany,  maple,  pear,  etc.  They  should  be  about  -J  of  an  inch  thick  in 
the  square  or  slightly  rounded  edges,  1  to  2%  inches  wide,  according  to  their 
length.  As  the  accuracy  of  a  drawing  depends  greatly  on  the  straightness  of 
the  lines,  the  edge  of  the  ruler  should  be  perfectly  straight.  To  test  this, 
place  a  sheet  of  paper  on  a  perfectly  smooth  board  ;  insert  two  very  fine 
needles  in  an  upright  position  through  the  paper  into  the  board,  distant  from 
each  other  nearly  the  length  of  the  ruler  to  be  tested ;  bring  the  edge  of  the 
ruler  against  these  needles,  and  draw  a  line  from  one  needle  to  the  other ; 
reverse  the  ruler,  bringing  the  same  edge  on  the  opposite  side  and  against 
the  needles,  and  again  draw  a  line.  If  the  two  lines  coincide,  the  edge  is 
straight ;  but,  if  they  disagree,  the  ruler  is  inaccurate,  and  must  be  re-jointed. 
When  one  ruler  has  been  tested,  the  other  can  be  examined  by  placing  their 
edges  against  the  correct  one,  and  holding  them  between  the  eye  and  the 
light. 

Triangles  may  be  made  of  the  same  kinds  of  wood  as  the  ruler,  and  some- 
what thinner,  and  of  various  sizes.  They  should  be  right-angled,  with  acute 
angles  of  45°,  or  of  60°  and  30°.  The  most  convenient  size  for  general  use 
measures  from  3  to  6  inches  on  the  side.  A  larger  size,  from  8  to  10  inches 
long  on  the  side,  is  convenient  for  making  drawings  to  a  large  scale.  Circular 
openings  are  made  in  the  body  of  the  triangle  for  the  insertion  of  the  end  of 
the  finger  to  give  facility  in  sliding  the  triangle  on  the  paper.  Triangles  are 
sometioies  made  as  large  as  15  to  18  inches  on  the  side  ;  but  in  this  case  they 
are  framed  in  three  pieces  of  about  1J  wide,  leaving  the  center  of  the  triangle 
open.  The  value  of  the  triangle  in  drawing  perpendicular  lines  depends  on 
the  accuracy  of  the  right  angle.  To  test  this  (Fig.  Ill),  draw  a  line  with  an 


DRAWING  INSTR 


41 


accurate  ruler  on  paper.  Place  the  right  angle  of  the  triangle  near  the  center 
of  this  line,  and  make  one  of  the  adjacent  sides  to  coincide  with  the  line  ;  now 
draw  a  line  along  the  other  adjacent  side,  which,  if  the  angle  is  strictly  a 
right  angle,  will  be  perpendicular  to  the  first  line.  Turn  the  triangle  on  this 
perpendicular  side,  bringing  it  into  the  posi- 
tion ABC';  if  now  the  sides  of  the  triangle 
agree  with  the  line  B  C'  and  A  B,  the  angle 
is  a  right  angle,  and  the  sides  straight.  If 
they  do  not  agree,  they  must  be  made  to  do 
so  with  a  plane,  if  right  angles  are  to  be 
drawn  by  the  triangle.  The  straightness 
of  the  hypothenuse  or  longest  side  can  be 
tested  like  a  common  ruler. 

The  T  square  is  a  thin  "  straight  edge  "  or  ruler,  a  (Fig.  112),  fitted  at  one 
end  with  a  stock,  b,  applied  transversely  at  right  angles.  The  stock  being  so 
formed  as  to  fit  and  slide  against  one  edge  of  the  drawing-board,  the  blade 
reaches  over  the  surface,  and  presents  an  edge  of  its  own  at  right  angles  to 


FIG.  111. 


FIG.  112. 

that  of  the  board,  by  which  parallel  straight  lines  may  be  drawn  upon  the 
paper.  The  stock  should  be  long  enough  to  give  sufficient  bearing  on  the 
edge  of  the  board,  and  heavy  enough  to  act  as  a  balance  to  the  blade,  and  to 
relieve  the  operation  of  handling  the  square.  The  blade  should  be  sunk  flush 
into  the  upper  half  of  the  stock  on  the  inside,  and  very  exactly  fitted.  It 
should  be  inserted  full  breadth,  as  shown  in  the  figure  ;  notching  and  dove- 
tailing is  a  mistake,  as  it  weakens  the  blade,  and  adds  nothing  to  the  secu- 
rity. The  upper  half  of  the  stock  should  be  about  \  inch  broader  than  the 
lower  half,  to  rest  firmly  on  the  board  and  secure  the  blade  lying  flatly  on  the 
paper. 

One  half  of  the  stock,  c  (Fig.  113),  is  in  some  cases  made  loose,  to  tarn 


FIG.  113. 


•upon  a  brass  swivel  to  any  angle  with  the  blade  a,  and  to  be  clenched  by  a 
screwed  nut  and  washer.     The  loose  stock  is  useful  for  drawing  parallel  lines 


42  DRAWING  INSTRUMENTS. 

obliquely  to  the  edges  of  the  board,  such  as  the  threads  of  screws,  oblique- 
columns,  and  connecting-roads  of  steam-engines. 

In  many  drawing-cases  will  be  found  the  parallel  ruler  (Fig.  114),  consist- 
ing of  two  rulers  connected  by  two  bars  moving  on  pivots,  and  so  adjusted 

that  the  rulers,  as  they  open, 

form  the  sides  of  a  parallelo- 

e^  G^  |         gram.     The  edge  of  one  of 

the  rulers  being  retained  in 
a  position  coinciding  with,  or 
parallel  to,  a  given  line,  the 
\  \>    I  other  ruler  may  be  moved, 

and   lines   drawn    along  its 
edge  must  also  be  parallel  to 

the  given  line.  This  instrument  is  only  useful  in  drawing  small  parallels,  and 
in  accuracy  and  convenience  does  not  compare  with  the  triangle  and  ruler,  or 
T  square. 

An  improvement  on  the  above  parallel  ruler  has  been  patented  by  Lieuten- 
ant-Commander Sigsbee,  U.  S.  N.  (Fig.  115),  in  which  the  blades  are* made 


FIG.  115. 

with  hinges,  by  which,  holding  one  blade  on  the  paper,  the  other  may  be  raised 
over  creases  or  torn  edges  of  the  paper,  or  over  thumb-tacks.  One  blade  can 
be  raised,  if  necessary,  at  right  angles  to  the  other,  still  preserving  the  parallel- 
ism of  lines  that  may  be  drawn  along  these  edges.  Small  cushions  of  rubber 
inserted  in  the  blades,  pressed  by  the  fingers,  prevent  the  slipping  of  the 
blades. 


FIG.  116. 


SWEEPS    AND    VARIABLE    CURVES. 


For  drawing  arcs  of  a  large  radius,  beyond  the  range  of  ordinary  com- 
passes, and  lines  not  circular  but  varying  in  curvature,  thin  slips  of  wood, 


DRAWING  INSTRUMENTS. 


FIG.  117. 


termed  sweeps  (Figs.  116  and  117),  are  usually  employed.     These  two  forms 

are  of  very  general  application,  but  others  of 

almost  every  form,  and  made  of  hard  rubber, 

can  be  purchased.    Whatever  be  the  nature  of 

the  curve,  some  portion  of  the  sweep  will  be 

found  to  coincide  with  its  commencement,  and 

it  can  be  continued   throughout  its  extent  by 

applying,  successively,  such  parts  of  the  sweep 

as  are  suitable,  care  being  taken  that  the  parts  are  tangent  to  each  other, 

and  that  the  continuity  is  not  injured  by  unskillful  junction. 

No  varnish  of  any  description  should  be  applied  to  any  of  the  wooden 
instruments  used  in  drawing,  as  the  best  varnish  will  retain  dust,  and  soil  the 
paper.  Use  the  wood  in  its  natural  state,  keeping  it  care- 
fully wiped.  Various  other  materials  besides  wood  have 
been  used,  as  steel  for  the  blades  of  the  T  square  and  the 
ruler  ;  the  objection  is  the  liability  to  soil  the  paper.  Glass 
is  frequently  used  for  the  ruler  and  the  triangle,  and  retains 
its  correctness  of  edge  and  angle,  but  it  is  too  heavy,  and 
liable,  of  course,  to  fracture. 

Thin  splines  are  also  to  be  had,  which,  held  in  position 
by  leaden  weights,  serve  admirably  for  a  guide  to  the  pen  in 
describing  curves  (Fig.  118).  For  the  same  purpose  a  thin, 
hard  rubber  ruler,  with  soft  rubber  backing,  answers  well, 
and,  as  it  can  be  readily  rolled  up,  is  extremely  portable. 

The  weights  above  shown  are  very  convenient  in  holding 
the  drawing-paper  on  the  board,  but  the  drawing-pins  (Fig. 
119),  steel  points,  or  tacks,  with  large,  flat  heads,  are  in 
general  use. 

Elliptic  and  parabolic  curves  are  furnished  in  sets,  but 
the  draughtsman  can  readily  make  a  model  out  of  thick 
card-board,  with  which  he  can  draw  a  very  uniform  curve. 

For  the  drawing  of  ellipses,  very  neat  trammels  or  com- 
passes, with  elliptic  guides  or  patterns,  may  be  purchased. 

The  drawing-pen  (Fig.  120)  is  used  for  drawing  straight  lines.  It  consists 
of  two  blades  with  steel  points  fixed  to  a  handle  ;  and  they  are  so  bent  that  a 
sufficient  cavity  is  left  between  them  for  the  ink,  when  the  ends 
of  the  steel  points  meet  close  together,  or  nearly  so.  The  blades 
are  set  with  the  points  more  or  less  open  by  means  of  a  mill- 
headed  screw,  so  as  to  draw  lines  of  any  required  fineness  or 
thickness.  One  of  the  blades  is  framed  with  a  joint,  so  that  by 
taking  out  the  screw  the  blades  may  be  completely  opened,  and 
the  points  effectively  cleaned  after  use.  The  ink  is  to  be  put 
between  the  blades  by  a  common  pen,  and  in  using  the  pen  it  should  be 
slightly  inclined  in  the  direction  of  the  line  to  be  drawn,  and  care  should  be 
taken  that  both  points  touch  the  paper  ;  and  these  observations  equally  apply 
to  the  pen-points  of  the  compasses  before  described.  The  drawing-pen  should 
be  kept  close  to  the  ruler  or  straight  edge,  and  in  the  same  direction  during 


FIG.  118. 


44 


DRAWING  INSTRUMENTS. 


the  whole  operation  of  drawing  the  line.     Care  must  be  taken  in  holding  the 

straight  edge  firmly  with  the  left  hand,  that  it  does  not  change  its  position. 

For  drawing  close  parallel  lines  in  mechanical  and 
architectural  drawings,  or  to  represent  canals  or  roads,  a 
double  pen  (Fig.  121)  is  frequently  used,  with  an  adjust- 
ing screw  to  set  the  pens  to  any  required  small  distance. 
This  is  usually  called  the  road-pen. 

Border-pens,  for  drawing  broad  lines,  are  double  pens 
with  an  intermediate  blade,  and  are  applicable  to  the 
drawing  of  map-borders.  The  same  work  may  be  done 
by  drawing  the  outer  lines  with  the  common  drawing-pen, 
and  filling  in  with  a  goose-quill,  cut  as  shown  in  Fig.  122. 
In  drawing  with  this  pen,  incline  the  drawing-board  so 
that  the  ink  will  follow  the  pen. 

The  curve-pen  (Fig.  123)  is  especially  designed  for  the 
ready  drawing  of  curved  lines. 

The  dotting-point  (Fig.  124)  resembles  a  drawing-pen, 
except  that  the  points  are  not  so  sharp.  On  the  back 
blade,  as  seen  in  the  engraving,  is  a  pivot,  on  which  may 
be  placed  a  dotting-wheel,  •«,  resembling  the  rowel  of  a 
spur ;  the  screw  ~b  is  for  opening  the  blades  to  remove  the 
wheel  for  cleaning  after  use,  or  replacing  it  with  one  of 
another  character  of  dot.  The  cap  c,  at  the  upper  end  of 
the  instrument,  is  a  box  containing  a  variety  of  dotting- 
wheels,  each  producing  a  different-shaped  dot.  These  are 
used  as  distinguishing  marks  for  different  classes  of  bound- 
aries on  maps  ;  for  instance,  one  kind  of  dot  distinguishes 
county  boundaries,  another  kind  town  boundaries,  a  third 

kind  distinguishes  that  which  is  both  a  county  and  a  town  boundary,  etc.,  etc. 

In  using  this  instrument,  the  ink  must  be  inserted  between  the  blades  above 


FIG.  120.        FIG.  121. 


FIG.  122. 

the  dotting-wheel,  so  that,  as  the  wheel  revolves,  the  points  shall  pass  through 
the  ink,  each  carrying  with  it  a  drop,  and  marking  the  paper  as  it  passes. 

It  sometimes  happens  that  the 
wheel  will  revolve  many  times 
before  it  begins  to  deposit  its 
ink  on  the  drawing,  thereby 
leaving  the  first  part  of  the  line 
altogether  blank,  and,  in  attempting  to  go  over  it  again,  the  first-made  dots 
are  liable  to  get  blotted.  This  evil  may  be  mostly  remedied  by  placing  a  piece 
of  blank  paper  over  the  drawing  to  the  very  point  the  dotted  line  is  to  com- 


FIG.  123. 


DRAWING  INSTRUMENTS. 


mence  at,  then  begin  with  drawing  the  wheel  over  the  blank  paper  first,  so 
that,  by  the  time  it  will  have  arrived  at  the  proper  point  of  commencement, 
the  ink  may  be  expected  to  flow  over  the  points  of  the  wheel,  and  make  the 
dotted  line  perfect  as  required. 

The  best  pricking-point  (Fig.  125)  is  a  fine  needle  held  in  a  pair  of  for- 
ceps, and  is  used  to  transfer  drawings  by  pricking  through  at  the  points  of  a 
drawing  into  the  paper  placed  beneath.  When  drawings  are  transferred  by 


I 


FIG.  124. 


FIG.  125. 


FIG.  126. 


tracing — a  prepared  black  sheet  being  placed  between  the  drawing  and  the 
paper  to  receive  the  tracing — the  eye-end  of  the  needle  forms  a  good  tracing- 
point. 

Compasses,  in  addition  to  pencil-points,  as  before  shown,  are  fitted  with 
movable  ink-points  and  lengthening  bars,  so  that  larger  circles  may  be  drawn. 
Compasses  should  have  joints  in  the  legs,  so  that  the  points,  pencil,  and  pen 
may  be  set  perpendicular  to  the  planes  in  which  the 
circles  are  described  (Fig.  126).  Compasses  of  this 
general  form  may  be  had  in  sizes  of  3£  to  7  inches. 

For  the  measurement  and  laying  off  of  small  spaces, 
and  the  describing  of  small  circles,  there  are  small  bow- 
compasses  (Fig.  127).  These  are  sometimes  made  with 
jointed  legs. 

For  the  measurement  or  laying  off  of  distances  the 
plain  dividers  are  convenient,  but  for  ready  and  close 
adjustment  the  hair  dividers  (Fig.  128)  are  most  suit- 
able. The  only  difference  is  that,  in  the  hair  dividers, 


FIG.  127. 


DRAWING  INSTRUMENTS. 


one  of  the  points  is  attached  to  the  body  by  a  spring,  and  by  means  of  the 
screw  b  it  can  be  moved  toward  or  from  the  fixed  point  a  very  small  amount 
more  accurately  than  by  closing  or  opening  the  dividers.  In  dividing  a  line 
into  equal  parts  especially,  it  enables  one  to  divide  the  excess  or 
deficit  readily. 

Large  screw  dividers  (Fig.  129)  are  used  for  the  same  purpose, 
but  they  belong  rather  to  the  mechanic  than  to  the  draughtsman. 
For  convenience  of  carrying  in  the  pocket,  there  are  portable 
or  turn-in  compasses  (Fig.  130). 


FIG.  128. 


FIG.  129. 


For  setting  off  very  long  lines,  or  describing  circles  of  large  radius,  learn 
compasses  are  used  (Fig.   131).      These  consist  of   a  mere    slip  of   wood,  A 


FIG.  130. 

which  is  readily  procured ;  two  brass  boxes,  B  and  0,  which  can  easily  be 
attached  to  the  beam,  and  connected  with  the  brass  boxes  are  the  two  points 
of  the  instrument,  G  and  H.  The  object  of  this  instrument  is  the  nice  adjust- 
ment of  the  points  G  and  H  at  any  definite  distance  apart ;  at  F  is  a  slow- 
motion  screw,  by  which  the  joint  G  may  be  moved  any  very  minute  quantity 
after  the  distance  from  F  to  G  has  been  adjusted  as  nicely  as  possible  by  the 
hand  alone.  The  important  parts  of  this  instrument  can  be  carried  in  a  very 
small  compass. 

There  are  beam  compasses  in  which  the  beam  is  graduated,  and  in  which 
the  boxes  corresponding  to  B  and  0,  in  Fig.  131,  are  fitted  with  vernier  or 
reading  plates,  to  afford  the  means  of  minutely  subdividing  the  divisions  on 
the  beam. 


DRAWING   INSTRUMENTS. 


47 


Proportional  dividers  (Fig.  132),  for  copying  and  reducing  drawings,  are 
found  in  most  cases  of  instruments. 

Closing  the  dividers  and  loosening  the  screw  0,  the  slide  may  be  moved  up 
in  the  groove  until  the  mark  on  the  slide  or  index  corresponds  with  the 
required  number;  then  clamping  the  screw,  the  space  inclosed 
between  the  long  points,  A  B,  will  be  as  many  times  that  between 
the  short  points,  E  D,  as  is  shown  by  the  number  opposite  the  in- 
dex. If  the  lines  are  to  be  reduced,  the  distances  are  measured 
with  the  long  points,  and  set  off  by  the  short  ones ;  if  the  lines 
.are  to  be  enlarged,  then  vice  versa. 

It  often  happens  that  the  length  of  the  points  becomes  re- 


0 


FKI.  131. 


FIG.  132. 


duced  by  use  or  accident,  and  the  division  on  the  instrument  then  becomes 
useless,  but  the  purpose  may  be  served  by  trial  on  paper,  moving  the  slide  up 
or  down  until  a  measured  line  is  reduced  or  enlarged,  as  required* 


SCALES. 

Practically,  a  two-foot  rule,  with  its  division  into  inches,  half  inch,  quarter 
inch,  eighth  inch,  and  sixteenth  inch,  may  be  made  use  of  as  a  scale  of  equal 
parts,  the  inch  or  any  of  its  parts  being  taken  as  the  unit  to  represent  a  foot, 
a  yard,  or  a  mile  ;  but  among  drawing  instruments,  scales  especially  adapted 
to  the  purpose  are  found  in  great  varieties  of  form,  division,  and  material. 

Fig.  133  represents  the  usual  scale  to  be  found  in  the  common  boxes  of 
drawing  instruments.  It  contains,  on  its  two  sides,  simply  divided  scales — a 
diagonal  scale  on  the  reverse  side  and  a  protractor  along  the  edges.  The 
simply  divided  scales  consist  of  a  series  of  equal  divisions  of  an  inch,  which 
are  numbered  1,  2,  3,  etc.,  beginning  from  the  second  division  on  the  left 
hand  ;  the  upper  part  of  the  left  division  in  each  is  subdivided  into  12  equal 
parts,  and  the  lower  part  into  10  equal  parts.  In  Fig.  134  the  scales  are 
marked  30,  35,  40,  etc.,  and  the  subdivisions  of  tenths  can  be  considered  as 
units,  one  mile,  or  one  chain,  or  one  foot,  then  each  primary  division  will 


48 


DRAWING  INSTRUMENTS. 


represent  ten  units,  ten  miles,  ten  chains,  or  ten  feet,  and  the  scale  is  said  to 
be  30,  35,  40  (according  to  the  scale  selected)  miles,  chains,  or  feet  to  the 

inch.  Thus,  suppose  that  it  were  required, 
on  a  scale  of  30  feet  to  the  inch,  to  lay  off 
47  feet.  On  the  scale  marked  30,  place  one 
point  of  the  compasses  or  dividers  at  4,  and 
bring  the  other  point  to  the  seventh  lower 
subdivisions,  counting  from  the  right,  and 
we  have  the  distance  required.  Each  of  the 
primary  divisions  may  be  regarded  as  unit, 
one  foot  for  instance ;  then  the  upper  sub- 
divisions are  twelfths  of  a  foot  or  inches,  and 
the  lower  subdivisions  tenths  of  an  inch. 

In  Fig.  133  the  scales  are  marked  at  the 
left,  1  inch,  f ,  £,  £ ;  the  primary  divisions 
are  1  inch,  f,  -J,  and  i  of  an  inch.  These 
scales  are  more  generally  used  for  drawings  of 
machinery  and  of  architecture,  while  those  of 
Fig.  134  are  for  topographical  drawings.  The 
applications  of  these  scales  are  similar  to  those 
already  described.  When  the  primary  divis- 
ions are  considered  inches,  then  the  drawings 
will  be  each  full,  f,  -J,  or  \  size,  according  to 
the  scale  adopted. 

On  the  selection  of  the  scale. — In  all  work- 
ing architectural  and  mechanical  drawings, 
use  as  large  a  scale  as  possible ;  neither  de- 
pend, even  in  that  case,  that  the  mechanics 
employed  in  the  construction  will  measure 
correctly,  but  write  in  the  dimensions  as  far 
as  practicable.  For  architectural  plans,  the 
scale  of  J-  an  inch  to  the  foot  is  one  of  very 
general  use,  and  convenient  for  the  mechanic, 
as  the  common  two-foot  rule  carried  by  all 
mechanics  is  subdivided  into  ^ths,  ^ths,  and 
sometimes  sixteenths  of  an  inch,  and  the  dis- 
tances on  a  drawing  to  this  scale  can  therefore 
be  easily  measured  by  them.  This  fact  should 
not  be  lost  sight  of  in  working  drawings. 
When  the  dimensions  are  not  written,  make 
use  of  such  scales  that  the  distances  may  be 
measured  by  the  subdivisions  of  the  common 

two-foot  rule  ;  thus,  in  a  scale  of  i  or  ^  full  size,  6  inches  or  3  inches  rep- 
resent one  foot ;  in  a  scale  of  an  inch  to  the  foot  or  twelfth  full  size,  each 
i  an  inch  represents  6  inches,  i  of  an  inch,  3  inches ;  but  when  •£  or  TV  an 
inch  to  the  foot,  or  any  similar  scale,  is  adopted,  it  is  evident  that  these 
divisions  can  not  be  taken  by  the  two-foot  rule.  The  scale  should  be  writ- 


FIG.  133. 


DRAWING  INSTRUMENTS. 


49 


ten  on  every  drawing,  or  the  scale  itself  should  be  drawn  on  the  margin. 
In  topographical  and  geodesic  drawings  the  latter  is  essential,  as  the  scale 
adopted  frequently  has  to  be  drawn  for  the  specific  purpose,  and  the  paper 


^ 

t 

t 

[ 

8 

Jo 

i  :  i, 

I 

j. 

i 

50 

;ip 

° 

i 

| 

i 

i 
I 

L 

-?  -,  L 

-1  - 

4 

; 

* 

it  5 

-i^p- 

i 

2 

£ 

i 

s      1       ,1 

I 

, 

it 

frfl 

"£p 

1, 

[ 

IG 

Is       1 

l\0 

[ 

35 

-^p 

5 

2 

j 

L 

1 

L 

30 

*  1  1  1 

^ 

» 

FIG.  134. 

itself  contracts  or  expands  with  every  atmospheric  change,  and  the  measure- 
ments will  therefore  not  agree  at  all  times  with  a  detached  scale ;  and,  more- 
over, a  drawing  laid  down  from  such  a  detached  scale,  of  wood  or  ivory,  will 
not  be  uniform  throughout,  for  on  a  damp  day  the  measurements  will  be  too 
short,  and  on  a  dry  day  too  long.  Mr.  Holtzapffel  has  sought  to  remedy  this 
inconvenience  by  the  introduction  of  paper  scales ;  but  all  kinds  of  paper  do 
not  contract  and  expand  equally,  and  the  error  is  therefore  only  partially  cor- 
rected by  his  ingenious  substitution  of  one  material  for  another. 


tn        1  ' 

1    2 

1     3 

4 

\     6 

1    6 

7 

|  963| 

X 

n 

yi 

Si     8 

tj    e 

9j     i 

S 

t 

01    I 

1  Sit  8;l 

fr  I  S 

m 

•III 

1  \  1 

1    1    1 

1    1    1 

1   1  1 

1 

FIG.  135. 

Plotting  scales  (Fig.  135)  are  scales  of  equal  parts,  with  the  divisions  on  a 
fiducial  edge,  by  which  any  length  may  be  marked  off  on  the  paper  without 
using  dividers.  There  are  also  small  offset  scales,  for  use  of  which  see  "  Topo- 
graphical Drawing." 

Sometimes  these  scales  are  made  with  edges  chamfered  on  both  sides,  and 
graduated  to  four  different  scales.  Sometimes  the  section  of  the  scale  is  tri- 
angular (Fig.  136),  with  six  scales  on  the  different  edges.  Both  of  these  scales 
are  convenient  as  portable  instruments.  To  avoid  the  objection  that  having 

A 


FIG.  136. 


many  scales  on  one  ruler  leads  the  draughtsman  into  error  by  the  confusion  of 
the  scales,  the  triangular  has  a  small  slip  of  metal,  A,  readily  put  on,  which 
covers  partially  the  scales  not  in  use. 


50 


DRAWING  INSTRUMENTS. 


To  divide  a  given  line  into  any  number  of  equal  parts  (Fig.  137). 
Let  A  B  be  the  line,  and  the  number  of  parts  be  ten.     Draw  a  perpendicu- 
lar at  one  extremity,  A,  of  the  line ;  with  a  plotting  scale  place  the  zero  at 

the  other  extremity,  B,  of  the  line ;  make  the  mark 
10  on  the  scale  coincide  with  the  perpendicular ; 
draw  a  line  along  the  edge  of  the  scale,  and  mark 
the  line  at  each  division  of  the  scale  1  to  9  ;  draw 
perpendiculars  through  these  marks  to  the  line  A  B, 
and  they  will  divide  A  B  into  ten  equal  parts. 

The  construction  is  based  on  the  principle  of 
the  proportions  of  parts  between  similar  triangles, 
and  it  is  evident  that  if  the  perpendicular  at  1  be 
taken  as  a  unit,  that  at  2  will  be  two  units,  and  so  on.  This  way  of  dividing 
a  line  will  often  be  found  convenient  in  practice.  The  lines  may  be  at  any 
angle  to  each  other,  and  the  lines  connecting  the  divisions  must  be  parallel  to 
the  line  completing  the  triangle.  The  above  figure  illustrates  the  construction 
of  diagonal  scales.  The  simply  divided  scales  give  only  two  denominations, 
primaries  and  tenths,  or  twelfths ;  but  more  minute  subdivision  is  attained  by 
the  diagonal  scale,  which  consists  of  a  number  of  primary  divisions,  one  of 
which  is  divided  into  tenths,  and  subdivided  into  hundredths  by  diagonal 
lines  (Fig.  138).  This  scale  is  constructed  in  the  following  manner :  Eleven 


FIG.  137. 


Fm.  138. 


parallel  lines  are  ruled,  inclosing  ten  equal  spaces ;  the  length  is  set  off  into 
equal  primary  divisions,  as  D  E,  E  1,  etc. ;  the  first  D  E  is  subdivided,  and 
diagonals  are  then  drawn  from  the  subdivisions  between  A  and  B,  to  those 


FIG.  139. 


between  D  and  E,  as  shown  in  the  diagram.     Hence  it  is  evident  that  at  every 
parallel  we  get  an  additional  tenth  of  the  subdivisions,  or  a  hundredth  of  the 


DRAWING  INSTRUMENTS; 


stlfli 


51 


primaries,  and  can  therefore  obtain  a  measurement  with  great  exactness  to 
three  places  of  figures.  To  take  a  measurement  of  (say)  168,  we  place  one  foot 
of  the  dividers  on  the  primary  1,  and  carry  it  down  to  the  ninth  parallel,  and 
then  extend  the  other  foot  to  the  intersection  of  the  diagonal,  which  falls 
from  the  subdivision  6,  with  the  parallel  that  measures  the  eight-hundredth 
part  (Fig.  139).  The  primaries  may,  of  course,  be  considered  as  yards,  feet,  or 
inches ;  and  the  subdivisions  as  tenths  and  hundredths  of  these  respective 
denominations. 

The  diagonals  may  be  applied  to  a  scale  where  only  one  subdivision  is 
required.     Thus,  if  seven  lines  be  (Fig.  140)  ruled,  inclosing  six  equal  spaces, 


7/V 

s/  V 

9/      -V- 

»7                   \2 

«/                        \1 

1                              \ 

0/2 

FIG.  140. 

and  the  length  be  divided  into  primaries,  as  A  B,  B  0,  etc.,  the  first  primary, 
A  B,  may  be  subdivided  into  twelfths  by  two  diagonals  running  from  6,  the 
middle  of  A  B,  to  12  and  0.  We  have  here  a  very  convenient  scale  of  feet  and 
inches.  From  C  to  6  is  1  foot  6  inches ;  and  from  C  on  the  several  parallels 
to  the  various  intersections  of  the  diagonals  we  obtain  1  foot  and  any  number 
of  inches  from  1  to  12. 

Vernier  scales  are  preferred  by  some  to  the  diagonal  scale  already  de- 
scribed. To  construct  a  vernier  scale  (Fig.  141)  by  which  a  number  to  three 
places  may  be  taken,  divide  all  the  primary  divisions  into  tenths,  and  number 


10 

2       4 

6       8 

f 

f           l           f           || 

I     I     I     I 

I     I    I     I 

I     I     I     I    I     I     I     I    I 

I    I     I    I     I    I    i    I     I    I     I    I     I     I    I     I    I     I    I     I 

._„  I 

I     I     I     I     I     I 

100  fr     a         6    J    4         2 

FIG.  141. 

these  subdivisions  1,  2,  3,  from  left  to  right.  Take  off  now  with  the  com- 
passes eleven  of  these  subdivisions,  set  the  extent  off  backward  from  the  end 
of  the  first  primary  division,  and  it  will  reach  beyond  the  beginning  of  this 
division,  or  zero  point,  a  distance  equal  to  one  of  the  subdivisions.  Now 
divide  the  extent  thus  set  off  into  ten  equal  parts,  marking  the  divisions  on 
the  opposite  side  of  the  divided  line  to  the  lines  marking  the  primary  divisions 
and  the  subdivisions,  and  number  them  1,  2,  3,  etc.,  backward  from  right  to 
left.  Then,  since  the  extent  of  eleven  subdivisions  has  been  divided  into  ten 
•equal  parts,  so  that  these  ten  parts  exceed  by  one  subdivision  the  extent  of  ten 
subdivisions,  each  one  of  these  equal  parts,  or,  as  it  may  be  called,  one  division 
of  the  vernier  scale,  exceeds  one  of  the  subdivisions  by  a  tenth  part  of  a  sub- 
division, or  a  hundredth  part  of  a  primary  division  ;  thus,  if  the  subdivision 
be  considered  10,  then  from  0  to  the  first  division  of  the  vernier  will  be  11 ;  to 
the  second,  22  ;  to  the  third,  33  ;  to  the  fourth,  44 ;  to  the  fifth,  55,  and  so 
on,  66,  77,  88,  99. 


52 


DRAWING  INSTRUMENTS. 


To  take  off  the  number  253  from  this  scale,  place  one  point  of  the  dividers 
at  the  third  division  of  the  vernier ;  if  the  other  point  be  brought  to  the  pri- 
mary division  2,  the  distance  embraced  by  the  dividers  will  be  233,  and  the 

dividers  must  be  extended  to  the  second  subdivision 
of  tenths  to  the  right  of  2.  If  the  number  were  213, 
then  the  dividers  would  have  to  be  closed  to  the  sec- 
ond subdivision  of  tenths  to  the  left  of  2.  To  take 
off  the  number  59  from  the  scale,  place  one  point  of 
the  dividers  at  the  ninth  division  of  the  vernier ;  if 
the  other  point  be  extended  to  the  0  mark,  the  di- 
viders will  embrace  99,  and  must  therefore  be  closed 
to  the  fourth  subdivision  to  the  left  of  0. 

These  numbers,  thus  taken,  may  be  253,  25 '3, 
2-53  ;  213,  21 -3,  2'13  ;  59,  5 -9,  .59,  according  as  the 
primary  divisions  are  taken  as  hundreds,  tens,  or 
units. 

The  construction  of  this  scale  is  similar  to  that 
of  the  verniers  of  theodolites  and  surveying  instru- 
ments ;  but,  in  its  application  to  drawing,  is  not  as 
simple  as  the  diagonal  scales  (Figs.  138,  140). 

The  sector  (Fig.  142),  now  seldom  used,  consists 
of  two  flat  rulers  united  by  a  central  joint,  and  open- 
ing like  a  pair  of  compasses.  It  carries  several  plain 
scales  on  its  faces,  but  its  most  important  lines  are 
in  the  pairs  or  double  scales,  running  accurately  to 
the  central  joint. 

The  principle  on  which  the  double  scales  are  con- 
structed is  that  similar  triangles  have  their  like  sides 
proportional  (Fig.  143).     Let  the 

^ ^C    lines  A  B,  AC,  represent  the  legs 

of  the  sector,  and  A  D,  A  E,  two 
equal  sections  from  the  center  ; 
then,  if  the  points  B  0  and  D  E 
be  connected,  the  lines  B  C  and 
D  E  will  be  parallel ;  therefore, 
the  triangles  A  B  C,  A  D  E,  will 
be  similar,  and,  consequently,  the 
sides  A  B,  B  C,  A  D,  D  E,  propor- 
tional— that  is,  as  A  B  :  B  C  :  : 
A  D  :  D  E  ;  so  that  if  A  D  be  the 
half,  third,  or  fourth  part  of  A  B, 
then  D  E  will  be  a  half,  third,  or 
fourth  part  of  B  C  ;  and  the  same 
holds  of  all  the  rest.  Hence,  if 
D  E  be  the  chord,  sine,  or  tangent 

of  any  arc,  or  of  any  number  of  degrees  to  the  radius  A  D,  then  B  C  will  be 
the  same  to  the  radius  A  B.     Thus,  at  every  opening  of  the  sector,  the  trans- 


DRAWING  INSTRUMENTS. 


53 


verse  distances  D  E  and  C  B  from  one  ruler  to  another  are  proportional  to 
the  lateral  distances,  measured  on  the  lines  A  B,  A  C.  It  is  to  be  observed 
that  all  measures  are  to  be  taken  from  the  inner  lines,  since  these  only  run 
accurately  to  the  center. 

On  the  scale  in  common  boxes  of  drawing  instruments,  the  edge  of  one 
side  is  divided  as  a  protractor,  for  the  laying  out  of  angles,  whose  use 
will  be  readily  understood  from  the  description  of  the  instrument,  when  by 
itself.  It  consists  of  a  semicircle  of  thin  metal  or  horn  (Fig.  144),  whose  cir- 
cumference is  divided  into  180  equal  parts  or  degrees  (180°).  In  the  larger 
protractors  each  of  these  divisions  is  subdivided. 

Application  of  the  protractor  (Fig.  144). — To  lay  off  a  given  angle  from  a 
given  point  on  a  straight  line,  let  the  straight  line  a  b  of  the  protractor  coin- 


cide with  the  given  line,  and  the  point  c  with  the  given  point ;  now  mark  on 
the  paper  against  the  division  on  the  periphery  coinciding  with  the  angle 
required  ;  remove  the  protractor,  and  draw  a  line  through  the  given  point  and 
the  mark. 

For  plotting  field-notes  expeditiously,  drawing  paper  can  be  obtained  with 
large,  full  circular  protractors  printed  thereon,  on  which  the  courses  can  be 
readily  marked,  and  thus  transferred  to  the  part  of  the  paper  required  by  a 
parallel  ruler,  or  by  triangle  and  ruler.  These  sheets  are  of  especial  use  in 
plotting  at  night  the  day's  work,  as,  on  account  of  the  large  size  of  protractor, 
angles  can  be  laid  off  with  greater  accuracy  than  by  the  usual  protractor  of 
a  drawing-instrument  case,  with  less  confusion  of  courses,  and  more  expe- 
ditiously. 

For  accurate  plotting  of  angles,  the  circular  protractor  (Fig.  145)  is  one  of 
the  best.  It  is  a  complete  circle,  A  A,  connected  with  its  center  by  four  radii, 
a  a  a  a.  The  center  is  left  open,  and  surrounded  by  a  concentric  ring  or  collar, 
&,  which  carries  two  radial  bars,  c  c.  To  the  extremity  of  one  bar  is  a  pinion, 
d,  working  in  a  toothed  rack  quite  round  the  outer  circumference  of  the  pro- 
tractor. To  the  opposite  extremity  of  the  other  bar,  c,  is  fixed  a  vernier, 
which  subdivides  the  primary  divisions  on  the  protractor  to  single  minutes, 


54  DRAWING  INSTRUMENTS. 

and  by  estimation  to  30  seconds.  This  vernier  is  carried  round  the  pro- 
tractor by  turning  the  pinion  d.  Upon  each  radial  bar,  c  c,  is  placed  a  branch, 
ee,  carrying  at  their  extremities  a  fine  steel  pricker,  whose  points  are  kept 
above  the  surface  of  the  paper  by  springs  placed  under  their  supports,  which 
give  way  when  the  branches  are  pressed  downward,  and  allow  the  points  to 


FIG.  145. 

make  the  necessary  punctures  in  the  paper.  The  branches  e  e  are  attached  to- 
the  bars  c  c  with  a  joint  which  admits  of  their  being  folded  backward  over 
the  instrument  when  not  in  use,  and  for  packing  in  its  case.  The  center  of 
the  instrument  is  represented  by  the  intersection  of  two  lines  drawn  at  right 
angles  to  each  other  on  a  piece  of  plate  glass,  which  enables  the  person  using 
it  to  place  it  so  that  the  center  or  intersection  of  the  cross-lines  may  coincide 
with  any  given  point  on  the  plan.  If  the  instrument  is  in  correct  order,  a  line 
connecting  the  fine  pricking  points  with  each  other  would  pass  through  the 
center  of  the  instrument,  as  denoted  by  the  before-mentioned  intersection  of 
the  cross-lines  upon  the  glass.  In  using  this  instrument,  the  vernier  should 
first  be  set  to  zero  (or  the  division  marked  360)  on  the  divided  limb,  and  then 
placed  on  the  paper,  so  that  the  two  fine  steel  points  may  be  on  the  given  line 
(from  whence  other  and  angular  lines  are  to  be  drawn),  and  the  center  of  the 
instrument  coincides  with  the  given  angular  point  on  such  line.  This  done, 
press  the  protractor  gently  down,  which  will  fix  it  in  position  by  means  of  very 
fine  points  on  the  under  side.  It  is  now  ready  to  lay  off  the  given  angle,  or  any 
number  of  angles  that  may  be  required,  which  is  done  by  turning  the  pinion  d 
till  the  opposite  vernier  reads  the  required  angle.  Then  press  downward  the 
branches  e  e,  which  will  cause  the  points  to  make  punctures  in  the  paper  at 
opposite  sides  of  the  circle  ;  which  being  afterward  connected,  the  line  will 
pass  through  the  given  angular  point,  if  the  instrument  was  first  correctly  set. 
In  this  manner,  at  one  setting  of  the  instrument,  a  great  number  of  angles  may 
be  laid  off  from  the  same  point. 

The  pantagraphs  are  used  for  the  copying  of  drawings  either  on  the  same 
scale,  on  a  reduced  scale,  or  on  an  enlarged  scale,  as  may  be  required.     The 


DRAWING  INSTRUMENTS. 


55 


form  of  pantagraph  as  shown  in  Fig.  146  consists  of  a  set  of  jointed  rulers, 
A,  B,  and  another,  C,  D,  about  one  half  the  length  of  the  former.  The  free 
ends  of  the  smaller  set  are  jointed  to  the  larger  at  about  the  center.  Casters 
are  placed  at  a  a,  etc.,  to  support  the  instrument  and  to  allow  an  easy  move- 


ment over  the  paper.  The  rulers  A  and  C  are  divided  with  a  scale  of  propor- 
tional parts,  marked  i,  -J,  etc.  These  arms  are  also  provided  with  movable 
indices,  E,  F,  which  can  be  fastened  at  any  division  by  clamp  screws.  Each 
index  is  provided  with  a  socket  adapted  to  carry  either  a  pencil  or  a  tracing 
point. 

Fig.  146  represents  the  instrument  in  the  act  of  reducing  the  plan  H  to  h, 
one  half  the  size.  The  tracing  point  is  placed  in  the  socket  at  E,  the  pencil  at 
F,  and  the  fulcrum  at  G.  The  indices,  E,  F,  are  clamped  each  at  £  on  the 
scales.  If  the  instrument  is  correct,  the  points  E,  F,  G,  are  in  a  straight  line. 
Pass  the  tracing  point  delicately  over  the  plan  H,  and  the  pencil  point  F  will 
trace  h,  one  half  the  original  size. 

If  the  object  had  been  to  enlarge  the  drawing  to  double  its  scale,  then  the 
tracer  must  have  been  placed  at  F,  and  the  pencil  at  E.  And  if  a  copy  be 
required,  retaining  the  scale  of  the  original,  then  the  slides  E  and  F  must  be 
placed  at  the  divisions  marked  1.  The  fulcrum  must  take  the  middle  sta- 
tion, and  the  pencil  and  tracer  those  on  the  exterior  rules  A  and  B  of  the 
instrument.  Another  form  of  this  instrument  is  shown  in  Fig.  147. 


FIG.  147. 

The  camera  lucida  is  sometimes  used  for  copying  and  reducing  topograph- 
ical drawings.  A  description  of  the  use  of  this  instrument  will  be  found  under 
the  head  of  topographical  drawing. 

The  drawing  table  and  drawing  board. — The  usual  size  of  the  drawing 
table  should  be  from  5  to  6  feet  long  and  3  feet  wide,  of  1|-  or  2-inch  white 
pine  plank  well  seasoned,  without  any  knots,  closely  joined,  glued,  doweled, 
and  clamped.  It  should  be  fixed  on  a  strong,  firm  frame  and  legs,  and  of  such 


56 


DRAWING  INSTRUMENTS. 


a  height  that  the  draughtsman,  as  he  stands  up,  may  not  have  to  stoop  to  his 
work.  The  table  is  usually  provided  with  a  shallow  drawer  to  hold  paper  or 
drawings.  Drawing  tables  are  made  portable  by  having  two  horses  for  their 
supports,  and  a  movable  drawing  board  for  the  top  ;  this  board  is  made  similar 
to  the  top  of  the  drawing  table,  but  of  inch  boards,  and  barred  at  the  ends. 
Various  woods  are  used  for  the  purposes,  but  white  pine  is  by  far  the  cheapest 
and  best.  Drawing  boards  should  be  made  truly  rectangular,  and  with  per- 
fectly straight  sides  for  the  use  of  the  T  square.  Two  sizes  are  sufficient  for 
common  purposes,  41  X  30  inches  to  carry  double  elephant  paper  with  a  mar- 
gin, and  31  X  24  inches  for  imperial  and  smaller  sizes.  Boards  smaller  than 
this  are  too  light  and  unsteady  in  handling. 

Small  boards  are  occasionally  made,  as  loose  panels  fitting  into  a  frame,  flush 
on  the  drawing  surface,  with  buttons  on  the  back  to  secure  them  in  position. 
The  panel  is  mostly  of  white  pine,  with  a  hard-wood  frame. 


DKAWIKG   PAPER. 

Hand-made  drawing  paper  is  usually  made  to  certain  standard  sizes  about 
as  follows  : 


Demy  ...........  20  inches  by 


inches. 


Medium  
Eoyal  

22| 
24 

'        17* 

'         191 

Super  Royal  
Imperial  

27i 
30 

j.  t/  ^ 

;     19^ 

'         22 

Elephant  

28 

'        23 

Columbier 35  inches  by  23^  inches. 

Atlas 34          "         26         " 

Double  Elephant.  40  27 

Antiquarian 53  '*          31 

Emperor 68  "         48         " 


But  of  late  machine-made  papers  are  the  most  used,  and  are  furnished  in 
rolls  of  widths  up  to  58  inches,  and  wider  can  be  obtained  by  order. 

Whatman's  white  paper  is  the  quality  most  usually  employed  for  finished 
drawings ;  it  will  bear  wetting  and  stretching  without  injury,  and,  when  so 
treated,  receives  color  readily.  For  ordinary  working  drawings,  where  damp- 
stretching  is  dispensed  with,  cartridge  paper,  in  rolls  of  a  coarser,  harder,  and 
tougher  quality,  is  preferable.  It  bears  the  use  of  India-rubber  better,  receives 
ink  on  the  original  undamped  surface  more  freely,  shows  a  fully  better  line, 
and,  as  it  does  not  absorb  very  rapidly,  tinting  lies  better  and  more  evenly 
upon  it.  For  delicate  small-scale  line-drawing,  the  thick  blue  paper,  such  as 
is  used  for  ledgers,  etc.,  imperial  size,  answers  exceedingly  well  ;  but  it  does 
not  bear  damp-stretching  without  injury,  and  should  be  merely  pinned  or 
waxed  down  to  the  board.  With  good  management,  there  is  no  ground  to  fear 
the  shifting  of  the  paper.  Good  letter  paper  receives  light  drawing  very  well ; 
of  course,  it  does  not  bear  much  fatigue. 

Drawings  destined  for  rough  usage  and  frequent  reference  should  be  on 
sheet  or  roll  drawing  paper,  backed  with  cotton  cloth,  which  can  be  purchased 
at  the  stationer's. 

Tracing  paper  is  a  preparation  of  tissue  paper,  transparent  and  qualified  to 
receive  ink  lines  and  tinting  without  spreading.  When  placed  over  a  drawing 
already  executed,  the  drawing  is  distinctly  visible  through  the  paper,  and  may 
be  copied  or  traced  directly  by  the  ink  instruments  ;  thus  an  accurate  copy  may 


DRAWING  INSTRUMENTS.  57 

be  made  with  great  expedition.  Tracings  may  be  folded  and  stowed  away  very 
conveniently  ;  but,  for  good  service,  they  should  be  mounted  on  cloth,  or  on 
paper  and  cloth,  with  paste. 

Tracing  paper  may  be  prepared  from  thick  tissue  paper  by  sponging  over 
one  surface  with  a  mixture  of  one  part  raw  linseed  oil  and  five  spirits  of  tur- 
pentine ;  five  gills  of  turpentine  and  one  of  oil  will  go  over  from  forty  to  fifty 
sheets  of  paper. 

Tracing  cloth  is  a  similar  preparation  of  linen,  and  is  preferable  for  its 
toughness  and  durability.  Tracing  paper  and  cloth  are  usually  to  be  had  in 
rolls,  and  tracings  on  cloth  are  now  preserved  as  originals,  and  copies  are  made 
from  them  by  some  sun  process. 

Mouth  Glue,  for  the  sticking  of  the  edges  of  drawing  paper  to  the  board,  is 
made  of  glue  and  sugar  or  molasses ;  it  melts  at  the  temperature  of  the  mouth, 
and  is  convenient  for  the  draughtsman. 

Drawing  paper  may  be  fixed  down  on  the  drawing  board  by  the  pins  at  the 
corners,  by  weights,  or  by  gluing  the  edges.  The  first  is  sufficient  when  110 
shading  or  coloring  is  to  be  applied,  and  if  the  sheet  is  not  to  be  a  very  long 
time  on  the  board  ;  and  it  has  the  advantage  of  preserving  the  paper  in  its 
natural  state.  For  shaded  or  tinted  drawings,  the  paper  must  be  damped  and 
glued  at  the  edges,  as  the  partial  wetting  of  paper,  loose  or  fixed  at  the  corners 
merely,  by  the  water-colors,  distorts  the  surface. 

Damp-stretching  is  done  as  follows  :  The  edges  of  the  paper  should  first  be 
cut  straight,  and,  as  near  as  possible,  at  right  angles  with  each  other  ;  also,  the 
sheet  should  be  so  much  larger  than  the  intended  drawing  and  its  margin  as 
to  admit  of  being  afterward  cut  from  the  board,  leaving  the  border  by  which  it 
is  attached  thereto  by  glue  or  paste,  as  we  shall  next  explain. 

The  paper  must  first  be  thoroughly  and  equally  damped  with  a  sponge  and 
clean  water,  on  the  opposite  side  from  that  on  which  the  drawing  is  to  be  made. 
When  the  paper  absorbs  the  Water,  which  may  be  seen  by  the  wetted  side  be- 
coming dim,  as  its  surface  is  viewed  slantwise  against  the  light,  it  is  to  be  laid 
on  the  drawing  board  with  the  wetted  side  downward,  and  placed  so  that  its 
edges  may  be  nearly  parallel  with  those  of  the  board  ;  otherwise,  in  using  a  J 
square,  an  inconvenience  may  be  experienced.  This  done,  lay  a  straight  flat 
ruler  on  the  paper,  with  its  edge  parallel  to,  and  about  half  an  inch  from,  one 
of  its  edges.  The  ruler  must  now  be  held  firm,  while  the  said  projecting  half- 
inch  of  paper  be  turned  up  along  its  edge  ;  then  a  piece  of  solid  or  mouth  glue, 
having  its  edge  partially  dissolved  by  holding  it  in  boiling  or  warm  water  for  a 
few  seconds,  must  be  passed  once  or  twice  along  the  turned-up  edge  of  the 
paper,  after  which,  by  sliding  the  ruler  over  the  glued  border,  it  will  be  again 
laid  flat,  and,  the  ruler  being  pressed  down  upon  it,  that  edge  of  the  paper  will 
adhere  to  the  board.  If  sufficient  glue  has  been  applied,  the  ruler  may  be  re- 
moved directly,  and  the  edge  finally  rubbed  down  by  an  ivory  book-knife,  or  by 
the  bows  of  a  common  key,  by  rubbing  on  a  slip  of  paper  placed  on  the  draw- 
ing paper,  so  that  the  surface  of  the  latter  may  not  be  soiled,  which  will  then 
firmly  cement  the  paper  to  the  board.  This  done,  another  but  adjoining  edge 
of  the  paper  must  be  acted  upon  in  like  manner,  and  then  the  remaining  edges 
in  succession  ;  we  say  the  adjoining  edges,  because  we  have  occasionally  ob- 


58  DRAWING  INSTRUMENTS. 

served  that,  when  the  opposite  and  parallel  edges  have  been  laid  down  first, 
without  continuing  the  process  progressively  round  the  board,  a  greater  degree 
of  care  is  required  to  prevent  undulations  in  the  paper  as  it  dries. 

Sometimes  strong  paste  is  used  instead  of  glue ;  but,  as  this  takes  a  longer 
time  to  set,  it  is  usual  to  wet  the  paper  also  on  the  upper  surface  to  within  an 
inch  of  the  paste  mark,  care  being  taken  not  to  rub  or  injure  the  surface  in  the 
process.  The  wetting  of  the  paper  in  either  case  is  done  for  the  purpose  of 
expanding  it ;  and  the  edges,  being  fixed  to  the  board  in  its  enlarged  state,  act 
as  stretchers  upon  the  paper,  while  it  contracts  in  drying,  which  it  should  be 
allowed  to  do  gradually.  All  creases  or  undulations  by  this  means  disappear 
from  the  surface,  and  it  forms  a  smooth  plane  to  receive  the  drawing. 

To  remove  the  paper  after  the  drawing  is  finished,  cut  oif  inside  the  pasted 
edge,  and  remove  the  edge  by  warm  water  and  the  knife. 

With  paneled  boards,  the  panel  is  taken  out,  and  the  frame  inverted  ;  the 
paper,  being  first  damped  on  the  back  with  a  sponge  slightly  charged  with 
water,  is  applied  equally  over  the  opening  to  leave  equal  margins,  and  is  pressed 
and  secured  into  its  seat  by  the  panel  and  bars. 

MOUNTING   PAPER   AND    DRAWINGS,    VARNISHING,   ETC. 

When  paper  of  the  requisite  quality  or  dimension  can  not  be  purchased 
already  backed,  it  may  be  mounted  011  cloth.  The  cloth  should  be  well 
stretched  upon  a  smooth  flat  surface,  being  damped  for  that  purpose,  and  its 
edges  glued  down,  as  was  recommended  in  stretching  drawing  paper.  Then 
with  a  brush  spread  strong  paste  upon  the  canvas,  beating  it  in  till  the  grain 
of  the  canvas  be  all  filled  up  ;  for  this,  when  dry,  will  prevent  the  canvas  from 
shrinking  when  subsequently  removed  ;  then,  having  cut  the  edges  of  the  paper 
straight,  paste  one  side  of  every  sheet,  and  lay  them  upon  the  canvas  sheet 
by  sheet,  overlapping  each  other  a  small  quantity.  If  the  drawing  paper  is 
strong,  it  is  best  to  let  every  sheet  lie  five  or  six  minutes  after  the  paste  is  put 
on  it,  for,  as  the  paste  soaks  in,  the  paper  will  stretch,  and  may  be  better  spread 
smooth  upon  the  canvas  ;  whereas,  if  it  be  laid  on  before  the  paste  has  moist- 
ened the  paper,  it  will  stretch  afterward  and  rise  in  blisters  when  laid  upon 
the  canvas.  The  paper  should  not  be  cut  off  from  its  extended  position  till 
thoroughly  dry,  which  should  not  be  hastened,  but  left  in  a  dry  room  to  do 
so  gradually,  if  time  permit  ;  if  not,  it  may  be  exposed  to  the  sun,  unless  in 
the  winter  season,  when  the  help  of  a  fire  is  necessary,  provided  it  is  not 
placed  too  near  a  scorching  heat. 

In  joining  two  sheets  of  paper  together  by  overlapping,  it  is  necessary,  in 
order  to  make  a  neat  joint,  to  feather-edge  each  sheet ;  this  is  done  by  care- 
fully cutting  with  a  knife  half  way  through  the  paper  near  the  edges,  and  on 
the  sides  which  are  to  overlap  each  other  ;  then  strip  off  a  feather-edged  slip 
from  each,  which,  if  done  dexterously,  will  form  a  very  neat  and  efficient  joint 
when  put  together. 

For  mounting  and  varnishing  drawings  or  prints,  stretch  a  piece  of  linen 
on  a  frame,  to  which  give  a  coat  of  isinglass  or  common  size,  paste  the  back  of 
drawing,  which  leave  to  soak,  and  then  lay  it  on  the  linen.  When  dry,  give  it 
at  least  four  coats  of  well-made  isinglass  size,  allowing  it  to  dry  between  each 


DRAWING   INSTRUMENTS.  59 

coat.     Take  Canada  balsam  diluted  with  the  best  oil  of  turpentine,  and  with  a 
clean  brush  give  it  a  full  flowing  coat. 

MANAGEMENT    OF   THE    INSTRUMENTS. 

In  constructing  preparatory  pencil-drawings,  it  is  advisable,  as  a  rule  of 
general  application,  to  make  no  more  lines  upon  the  paper  than  are  necessary 
to  the  completion  of  the  drawing  in  ink  ;  and  also  to  make  these  lines  just  so 
dark  as  is  consistent  with  the  distinctness  of  the  work.  With  respect  to  the 
first  idea,  it  is  of  frequent  application  :  in  the  case,  for  example,  of  the  teeth 
of  spur  wheels,  where,  in  many  instances,  all  that  is  necessary  to  the  drawing 
of  their  end  view  in  ink  are  three  circles,  one  of  them  for  the  pitch  line,  and 
the  two  others  for  the  tops  and  bottoms  of  the  teeth  ;  and  again,  to  draw  the 
face  view  of  the  teeth — that  is,  in  the  edge  view  of  the  wheel — we  have  only 
to  mark  off  by  dividers  the  positions  of  the  lines  which  compose  the  teeth,  and 
draw  four  pencil  lines  for  the  two  sides,  and  the  top  and  bottom  of  the  eleva- 
tion. And  here  we  may  remark  the  inconvenience  of  that  arbitrary  rule,  by 
which  it  is  by  some  insisted  that  the  pupil  should  lay  down  in  pencil  every  line 
that  is  to  be  drawn  before  finishing  it  in  ink.  It  is  often  beneficial  to  ink  in 
one  part  of  a  drawing  before  touching  other  parts  at  all ;  it  prevents  confusion, 
makes  the  first  part  of  easy  reference,  and  allows  of  its  being  better  done,  as  the 
surface  of  the  paper  inevitably  contracts  dust  and  becomes  otherwise  soiled  in 
the  course  of  time,  and  therefore  the  sooner  it  is  done  with  the  better. 

Circles  and  circular  arcs  should,  in  general,  be  inked  in  before  straight  lines, 
as  the  latter  may  be  more  readily  drawn  to  join  the  former  than  the  former 
the  latter.  When  a  number  of  circles  are  to  be  described  from  one  center,  the 
smaller  should  be  inked  first,  while  the  center  is  in  better  condition.  When  a 
center  is  required  to  bear  some  fatigue,  it  should  be  protected  with  a  thickness- 
of  stout  card  glued  or  pasted  over  it,  to  receive  the  compass-leg. 

India-rubber  is  the  ordinary  medium  for  cleaning  a  drawing,  and  for  cor- 
recting errors  in  the  pencil.  For  slight  work  it  is  quite  suitable ;  that  sub- 
stance, however,  operates  to  destroy  the  surface  of  the  paper  ;  and,  by  repeated 
application,  it  so  ruffles  the  surface,  and  imparts  an  unctuosity  to  it,  as  to  spoil 
it  for  fine  drawing,  especially  if  ink  shading  or  coloring  is  to  be  applied.  It  is 
much  better  to  leave  trivial  errors  alone,  if  corrections  by  the  pencil  may  be 
made  alongside  without  confusion,  as  it  is,  in  such  a  case,  time  enough  to 
clear  away  superfluous  lines  when  the  inking  is  finished. 

For  cleaning  a  drawing,  a  piece  of  bread  two  days  old  is  preferable  to  India- 
rubber,  as  it  cleans  the  surface  well  and  does  not  injure  it.  When  ink  lines  to 
any  considerable  extent  have  to  be  erased,  a  small  piece  of  damped  soft  sponge 
may  be  rubbed  over  them  till  they  disappear.  As,  however,  this  process  is  apt 
to  discolor  the  paper,  the  .sponge  must  be  passed  through  clean  water,  and  ap- 
plied again  to  take  up  the  straggling  ink.  For  ordinary  small  erasures  of  ink 
lines,  a  sharp  rounded  pen-blade,  applied  lightly  and  rapidly,  does  well,  and  the 
surface  may  be  smoothed  down  by  the  thumb-nail.  In  ordinary  working  draw- 
ings, a  line  may  readily  be  taken  out  by  damping  it  with  a  hair-pencil  and 
quickly  applying  the  India-rubber  ;  and  to  smooth  the  surface  so  roughened,  a 
light  application  of  the  knife  is  expedient.  In  drawings  intended  to  be  highly 


60  DRAWING  INSTRUMENTS. 

finished,  particular  pains  should  be  taken  to  avoid  the  necessity  for  corrections, 
as  everything  of  this  kind  detracts  from  the  appearance. 

In  using  the  square,  the  more  convenient  way  is  to  draw  the  lines  off  the 
left  edge  with  the  right  hand,  holding  the  stock  steadily  but  not  very  tightly 
against  the  edge  of  the  board  with  the  left  hand.  The  convenience  of  the  left 
edge  for  drawing  by  is  obvious,  as  we  are  able  to  use  the  arms  more  freely,  and 
we  see  exactly  what  we  are  doing. 

To  draw  lines  in  ink  with  the  least  amount  of  trouble  to  himself,  the  me- 
chanical draughtsman  ought  to  take  the  greater  amount  of  trouble  with  his 
tools.  If  they  be  well  made,  and  of  good  stuff  originally,  they  ought  to  last 
through  three  generations  of  draughtsmen  ;  their  working  parts  should  be  care- 
fully preserved  from  injury,  they  should  be  kept  well  set,  and,  above  all,  scru- 
pulously clean.  The  setting  of  instruments  is  a  matter  of  some  nicety,  for 
which  purpose  a  small  oil-stone  is  convenient.  To  dress  up  the  tips  of  the 
blades  of  the  pen  or  of  the  bows,  as  they  are  usually  worn  unequally  by  the 
customary  usage,  they  may  be  screwed  up  into  contact  in  the  first  place,  and 
passed  along  the  stone,  turning,  upon  the  point  in  a  directly  perpendicular 
plane,  till  they  acquire  an  identical  profile.  Being  next  unscrewed  and  exam- 
ined to  ascertain  the  parts  of  unequal  thickness  round  the  nib,  the  blades  are 
laid  separately  upon  their  backs  on  the  stone,  and  rubbed  down  at  the  points, 
till  they  be  brought  up  to  an  edge  of  uniform  fineness.  It  is  well  to  screw 
them  together  again,  and  to  pass  them  over  the  stone  once  or  twice  more,  to 
bring  up  any  fault ;  to  retouch  them  also  on  the  outer  and  inner  side  of  each 
blade,  to  remove  barbs  or  fraying  ;  and,  finally,  to  draw  them  across  the  palm 
of  the  hand. 

The  China  ink  which  is  commonly  used  for  line-drawing  ought  to  be 
rubbed  down  in  water  to  a  certain  degree,  avoiding  the  sloppy  aspect  of  light 
lining  in  drawings,  and  making  the  ink  just  so  thick  as  to  run  freely  from  the 
pen.  This  medium  degree  may  be  judged  of  after  a  little  practice  by  the  ap- 
pearance of  the  ink  on  the  palette.  The  best  quality  of  ink  has  a  soft  feel  when 
wetted  and  smoothed  ;  free  from  grit  or  sediment,  and  musky.  The  rubbing 
of  China  ink  in  water  tends  to  crack  and  break  away  the  surface  at  the  point  ; 
this  may  be  prevented  by  shifting  at  intervals  the  position  of  the  stick  in  the 
hand  while  being  rubbed,  and  thus  rounding  the  surface.  Nor  is  it  advisable, 
for  the  same  reason,  to  bear  very  hard,  as  the  mixture  is  otherwise  more  evenly 
made,  and  the  enamel  of  the  palette  is  less  rapidly  worn  off.  When  the  ink,  on 
being  rubbed  down,  is  likely  to  be  for  some  time  required,  a  considerable  quan- 
tity of  it  should  be  prepared,  as  the  water  continually  vaporizes ;  it  will  thus 
continue  for  a  longer  time  in  a  condition  fit  for  application.  The  pen  should 
be  leveled  in  the  ink,  to  take  up  a  sufficient  charge  ;  and,  to  induce  the  ink  to 
enter  the  pen  freely,  the  blades  should  be  lightly  breathed  upon  before  immer- 
sion. After  each  application  of  ink,  the  outsides  of  the  blades  should  be 
cleaned,  to  prevent  any  deposit  of  ink  upon  the  edge  of  the  squares. 

To  keep  the  blades  of  his  inkers  clean  is  the  first  duty  of  a  draughtsman 
who  is  to  make  a  good  piece  of  work.  Pieces  of  blotting  or  unsized  paper  and 
cotton  velvet,  wash-leather,  or  even  the  sleeve  of  a  coat,  should  always  be  at 
hand  while  a  drawing  is  being  inked.  When  a  small  piece  of  blotting  paper  is 


DRAWING  INSTRUMENTS.  61 

folded  twice  so  as  to  present  a  corner,  it  may  usefully  be  passed  between  the 
blades  of  the  pen  now  and  then,  as  the  ink  is  liable  to  deposit  at  the  point  and 
obstruct  the  passage,  particularly  in  fine  lining  ;  and  for  this  purpose  the  pen 
must  be  unscrewed  to  admit  the  paper.  But  this  process  may  be  delayed  by 
drawing  the  point  of  the  pen  over  a  piece  of  velvet,  or  even  over  the  surface  of 
thick  blotting-paper  ;  either  method  clears  the  point  for  a  time.  As  soon  as 
any  obstruction  takes  place,  the  pen  should  be  immediately  cleaned,  as  the 
trouble  thus  taken  will  always  improve  and  expedite  the  work.  If  the  pen 
should  be  laid  down  for  a  short  time  with  the  ink  in  it,  it  should  be  unscrewed 
to  keep  the  points  apart,  and  so  prevent  deposit  ;  and,  when  done  with  alto- 
gether for  the  occasion,  it  ought  to  be  thoroughly  cleaned  at  the  nibs.  This 
will  preserve  its  edges  and  prevent  rusting. 

For  the  designing  of  machinery,  it  is  very  convenient  to  have  some  scale  of 
reference  by  which  to  proportion  the  parts  ;  for  this  purpose  a  vertical  and 
horizontal  scale  may  be  drawn  on  the  walls  of  the  room. 


EXERCISES  WITH  THE 

Before  proceeding  to  the  construction  of  finished  drawings,  skill  should  be 
acquired  in  the  use  of  the  drawing-pen,  supplemented  often  by  the  steel  pen. 
Beginning  with  lines,  outlines  of  figures,  alphabets,  and  the  like,  the  draughts- 
man should  strive  to  acquire  the  habit  of  readily  drawing  clean,  uniform  lines, 
without  abruptness  or  breaks,  where  straight  lines  connect  with  curved  ones. 
Draw  straight  lines  of  different  grades  : 

as,          fine  - 

medium  -  -  —  •  --  • 
coarse  ^  —  —  ^^—  —  —  —  —  —  — 

at  first,  lines  of  indefinite  length,  taking  care  that  they  are  drawn  perfectly 
straight  and  of  uniform  width  or  grade  ;  then  draw  lines  of  definite  length 
between  assumed  points,  taking  care  to  terminate  the  lines  exactly  at  these 
points.  Lines  as  above  are  full  lines,  the  grades  depending  on  the  effect  which 
the  draughtsman  wishes  to  give. 

Draw  dotted  lines,  broken  lines,  and  broken  and  dotted  lines,  of  different 
grades  : 


Draw  fine  lines  at  uniform  distances  from  each  other 


DRAWING  INSTRUMENTS. 


To  give  uniform  appearance,  the  lines  must  be  of  uniform  grade  and  equally 
spaced.  Practice  in  lines  of  this  sort  is  important,  as  they  are  much  used  in 
drawing  to  represent  sections,  shades,  and  conditions,  as  soundings  on  charts, 
density  or  characteristics  of  population,  areas  of  rain,  temperature,  and  the  like. 
Draw  lines  as  in  Fig.  148.  These  lines  are  diagonal  with  the  border-lines,  and 


FIG.  148. 


are  used  to  represent  sections  of  materials.     In  the  figure,  lines  differently  in- 
clined represent  different  pieces  of  the  same  material. 

Sections  of  different  materials  may  be  represented  in  different  kinds  of 
lines,  as  in  Figs.  149,  150,  151. 


FIG.  149. 


FIG.  150. 


FIG.  151. 


These  particular  ones  are  used  to  represent  sections  of  wrought-iron,  steel, 
and  cast-iron  ;  but  they  may  be  used  to  represent  different  colors,  the  location 
of  different  mineral  or  agricultural  products,  etc. 

To  represent  cylindrical  surfaces  (Fig.  152). 

Draw  a  semi-circumference,  and  mark  on  it  a  number  of  points,  at  equal 
distances  apart,  and  through  these  points  draw  lines  perpendicular  to  the 


FIG.  152. 


FIG.  153. 


diameter  across  the  surface  to  be  represented.  It  is  not  absolutely  necessary 
that  the  central  space  should  be  equal  to  the  others ;  it  will  be  more  effective 
to  leave  out  two  of  the  lines,  and  make  it  to  this  extent  wider. 


DRAWING  INSTRUMENTS. 


63 


To  construct  a  mass  of  equal  squares  (Fig.  153). 

Lay  off  a  right  angle,  and  on  its  sides  mark  as  many  points,  at  equal  dis- 
tances apart,  as  may  be  necessary  ;  through  these  points  draw  lines  parallel  to 
the  sides. 

Or,  construct  a  rectangle  ;  mark  on  its  sides  as  many 
points,  at  equal  distances  apart,  as  may  be  necessary ; 
through  these  points  draw  the  lines. 

To  construct  the  squares  diagonally  to  the  base  (Fig. 
154). 

Mark  on  the  sides  of  the  right  angle  as  many  points, 
at  distances  apart  equal  to  the  diagonal  of  the  required 
squares,  as  may  be  necessary.   Con- 
nect these  points  by  lines  as  shown, 
and  through  the  same  points  draw 
lines  at  right  angles  to  the  others. 

Or,  as  above,  construct  a  rec- 
tangle, and  mark  on  its  sides  points 
at  distances  apart  equal  to  the  di- 
agonal of  the  required  squares. 

To  cover  a  surface  with  equi-  FlG  154 

lateral  triangles  (Fig.  155). 

Construct  an  angle  of  60°,  and  mark  on  its  sides  points  at  distances  apart 
equal  to  the  side  of  the  triangle.  Connect  these  points ;  and  through  these 
points  draw  lines  parallel  to  the  sides  of  the  angle. 

Figures  composed  of  two  triangles,  with  the  same  base,  are  called  lozenges. 
Six  triangles  may  be  arranged  as 
a  hexagon.      The  whole  surface 
may  be  arranged  in  lozenges  or 
hexagons. 

To  cover  a  surface  with  octa- 
gons and  squares  (Fig.  156). 

Lay  off  the  surface  in  squares 
having  sides  equal  to  the  width 
of  the  octagons.  Corner  the  outer 
squares  to  form  octagons,  as  by 
Prob.  XL.,  page  21.  Extend  the 
sides  of  these  octagons  across  the 
other  squares,  and  similar  corners 
will  be  cut  off,  and  the  octagons 
and  squares  required  will  be  com-  FlG  155 

plete. 

With  the  aid  of  paper  thus  covered  with  squares,  triangles,  and  lozenges, 
various  geometrical  designs  may  be  readily  constructed,  pleasing  in  their  effect, 
and  affording  good  practice  to  young  draughtsmen. 

In  the  examples  given  of  designs  constructed  on  squares  or  triangles,  if  it  is 
desired  to  increase  or  diminish  the  size  of  the  original  designs,  it  is  only  neces- 
sary to  make  the  sides  of  the  squares  or  triangles  larger  or  smaller,  and  taking 


64  DRAWING  INSTRUMENTS. 

relatively  the  same  squares  for  the  construction  of  the  figures.  In  transferring 
designs  and  drawings  from  books  or  plates,  on  which  squares  can  not  be  drawn, 
it  is  very  convenient  to  have  a  square  of  glass,  with  squares  upon  it,  which  may 
be  laid  on  the  drawing,  and  thus  serve  the  same  purpose  as  if  squares  had  been 

C 

F 


drawn.  The  glass  may  be  readily  prepared  by  painting  one  of  its  surfaces  with 
a  thin  coat  of  gum,  and  drawing  squares  upon  it  with  the  drawing-pen  ;  if 
every  fifth  or  tenth  line  be  made  fuller  or  in  a  different  color,  it  will  be  still 
more  convenient  for  reference. 

Fig.  157  is  the  front  view  and  side  of  an  acanthus-leaf,  of  which  the  sur- 
faces are  covered  with  squares,  somewhat  larger  than  would  be  recommended 


FIG.  157. 


FIG.  158. 


in  practice,  but  sufficient  to  illustrate  the  principle,  which  may  be  done  by 
the  learner  on  the  same  or  other  sized  squares.  If  the  same  size,  the  intersec- 
tions of  the  lines  of  the  figure  with  those  of  the  squares  are  easiest  transferred 
by  a  straight-edged  slip  of  paper,  placed  along  a  line,  and  making  all  the  inter- 
sections at  once,  and  then  transferring  the  marks  to  the  copy. 


DRAWING  INSTRU 


65 


Fig.  158  is  the  side-view  of  the  acanthus-le^f,  in  a  reversed  position  from 
the  original  (Fig.  157)  ;  that  is,  right-handed,  while  the  original  is  left-handed. 
It  will  readily  be  understood  how  this  may  be  done  by  observing  the  letters  on 
the  side  and  the  numerals  at  the  top  of  the  squares. 

Fig.  159  represents  the  construction  of  Gothic  letters  and  numerals  on  a 
system  of  squares.  These  letters  are  formed  mechanically  by  the  drawing-pen 
and  dividers. 

Fig.  160  are  Italic  letters,  drawn  on  rhombs,  in  which  the  upright  lines 
are  inclined  to  horizontal. 

On  pages  66,  67,  68,  69,  are  specimens  of  type  taken  from  the  printer's 
font,  which  can  be  readily  transferred  to  a  drawing,  by  covering  them  with 
a  bit  of  glass  or  horn,  laid  off  in  squares,  as  described  above.  Printers'  let- 
ters are  in  general  well  proportioned,  but  it  is  customary  often  to  distort 
letters,  to  call  attention  to  them,  or  to  adapt  them  to  the  position  in  which 
they  are  to  be  placed.  Spaces  between  the  letters  are  in  printing  uniform, 
but  in  drawing,  when  such  letters  come  together  as  F  and  A,  L  and  T,  one 
wide  at  top  and  the  other  at  bottom,  the  spacing  between  them  may  be 
reduced  a  little.  The  acquisition  of  a  ready  hand  in  lettering  enables  a 
draughtsman  to  give  a  finish  to  a  good  drawing  or  map  which  might  other- 
wise be  spoiled  by  poor  lettering. 


FIG.  159. 


66  DRAWING  INSTRUMENTS. 


LARGE     ROMAN. 


ABC  DE  FGH   IJ 
KLMN  OP  QRST 

TJV  WX  YZ 


SMALL    EOMAK'. 


abc  de  fgh  ij  klmn  op 

qrst  uv  wx  yz 

1234567890 


DRAWING  INSTRUMENTS.  67 

ENGLISH    GOTHIC. 

ABC  DE  FGH  IJ  KLMN  OP 

QRST  UV  WX  YZ 

1234567890 


ITALIC. 


ABC   DE    FGH   IJ   KLMN    OP    QRST 

UV    WX     YZ 

abc   de  fgli   ij  klmn  op  qrst  uv  wx  yz 


TUSCAN. 


ABC  DE  FGH  IJ  KLMN  OP  QRST 

UV  WX  YZ 
1234567890 

ABC  DE  FGH  IJ 
ELMNOPQEST 

U7WZYZ 

abc  de  fgh  ij  klmn  op 

qist  uv  wx  yz 


68 


DRAWING  INSTRUMENTS. 

TELEGRAPH. 


OEK AMEKTED. 


UV  WX 


de 


uv  wx 


OLD    ENGLISH. 


Jfi 


QV  CD 


1 


e;Ci- 


ak  to  fg|  ij  klrnn  0p  qrst 


DRAWING  INSTRUMENTS.  69 

ENGLISH    CHURCH    TEXT. 


aic 

t  § 

aht  it  fgji  ij  klmn  up  qtst  un  rax 


MEDIEVAL 


13 


«Jf 

afir  bp  fg|  ij  hlmn  op  qrsf-  uti  tof 

BI  F&       II 


ij   felmtx   xxp   qrst   utr 


Paper  printed  in  squares  is  used  by  designers  of  figures  for  calicoes,  silks, 
and  woolens.  For  the  engineer,  there  is  a  class  of  papers  called  cross-section 
papers,  sold  in  sheets  or  rolls,  and  of  various  scales,  originally  intended,  as  the 
name  implies,  for  cross-sections  of  railway  or  canal  cuts,  but  now  extensively 
employed  by  the  architectural  and  mechanical  designer  for  the  rough  sketches 
of  works  either  executed  or  to  be  executed  ;  by  the  sanitarian  for  the  plotting  of 
death-rates  ;  for  thermometric  and  hygrometric  readings  ;  by  the  broker  and 
merchant  for  the  graphic  representation  of  the  prices  of  gold,  stocks,  or  articles 
of  merchandise,  during  a  term  of  years  ;  by  the  railway  superintendent  for  the 
movement  of  trains  ;  and  for  multitudes  of  other  uses.  These  may  hardly  be 
considered  in  the  light  of  drawings ;  but,  as  they  involve  the  drawing  of  lines, 
shading  of  spaces,  and  lettering,  and  as  there  is  no  head  of  drawing  under 


70 


DRAWING  INSTRUMENTS. 


which  this  use  of  cross-section  paper  can  be  classed,  it  seems  proper  to  give 
here  a  few  illustrations,  which  will  show  its  general  application. 

Fig.  161  shows  a  graphical  method  of  determining  the  equivalent  values  of 
the  metric  system  of  measurements  in  United  States  units,  or  vice  versa.  The 
vertical  scale  represents  the  metric  units,  and  the  horizontal  the  common  or 


oo  TJI  to  so 

UNITED  STATES  UNITS. 
FIG.  161. 


United  States  units.  The  method  of  using  the  diagram  can  be  best  shown  by 
taking  one  or  two  examples. 

What  is  the  equivalent  value  of  seven  kilometres  in  miles  ?  Read  upward 
on  the  metric  scale  to  7,  then  read  on  that  horizontal  line  to  the  point  of  in- 
tersection with  the  line  designated  "MILES  &  KILOMETRES,"  that  is,  at 
the  point  on  the  United  States  scale  of  units  representing  4 '35  ;  therefore, 
seven  kilometres  are  equal  to  4*35  miles. 

What  is  the  value  of  five  pounds  in  kilogrammes  ?  The  process  is  the  same 
as  the  foregoing,  except  that,  to  change  United  States  units  into  the  metric 
units,  first  read  horizontally,  then  upward.  The  result  will  be  in  this  case  that 
five  pounds  is  found  equal  to  2*25  kilogrammes.  The  divisions  may  represent 
single  units,  ten  units,  one  hundred  units,  etc.  ;  that  is,  if  we  had  wished  to 
find  the  equivalent  of  500  pounds,  it  would  have  been  225  kilogrammes. 


DRAWING   INSTRUMENTS. 


71 


Fig.  162  is  a  diagram  illustrating  graphically  the  difference  charged  on  a  ton 
of  merchandise  per  mile}  on  the  New  York  Central  and  Hudson  River  Railroad 
and  the  Erie  Canal,  for  every  year  between  185?  and  1880 ;  the  values  being 


FIG.  162. 

published  in  the  Report  of  the  United  States  Bureau  of  Statistics  for  1880. 
The  higher  values  in  every  case  represent  the  railroad  rates  and  the  lower  the 
canal  rates.  The  black  band  shows  the  difference  between  these  values.  In 
1865,  for  instance,  the  railroad  rates  were  3 '30  cents,  and  the  canal  1'02  cents, 
the  difference  being  2  -28  cents. 

Fig.  163  is  made  up  from  the  time-table  of  the  New  York,  New  Haven,  and 
Hartford  Railroad,  showing  the  movement  of  trains,  two  from  New  York  and 
two  from  New  Haven,  the  abscissas  (horizontal  lines)  being  cut  off  on  a  scale 
of  miles  for  each  station,  the  ordinates  (vertical  lines)  being  a  scale  of  hours. 


DRAWING  INSTRUMENTS. 


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Fig.  164  shows  the  method  of  finding  the  average  of  a  number  of  observa- 
tions.    The  figure  represents  the  path  of  a  float  in  a  wooden  flume  or  channel, 


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taken  from  the  last  edition  of  Francis's  "Lowell  Hydraulic  Experiments." 
The  cut  was  copied  directly  on  the  wood,  and  is  therefore  reversed.     The 


DRAWING  INSTRUMENTS.  73 

width  of  the  cut  represents  the  width  of  the  flume,  each  abscissa  being  one 
foot ;  the  ordinates  are  the  speeds  of  float  in  divisions  of  0*1  foot  per  second  ; 
the  o  o  on  the  cut  are  meant  to  represent  the  floats  in  their  observed  path  and 
speed  ;  and  the  curved  line  the  average  velocity  in  the  different  threads  of  the 
stream. 

Fig.  165  is  from  Clarke's  "  Railway  Machinery."  The  abscissas  represent 
the  speed  in  miles  per  hour  ;  the  ordinates  the  pounds  per  ton  resistance  of  a 
100- ton  train. 


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FIG.  165, 

Fig.  166  is  a  diagram  illustrating  the  daily  mortality  during  the  month  of 
November,  1873,  in  New  York  City.  The  figure  is  a  copy  of  a  portion  of  the 
chart  published  in  the  Report  of  the  Metropolitan  Board  of  Health  for  that 
year.  The  lower  irregular  line  shows  the  daily  mortality.  The  upper  single 
irregular  line  shows  the  daily  average  temperature.  The  terminal  cross-lines  at 
the  ends  of  perpendicular  bars  show  the  daily  range  of  temperature.  The 
double  irregular  line  shows  the  daily  humidity,  saturation  being  100  on  the 
scale  of  temperatures.  The  black  bands  in  the  upper  portion  of  the  diagram 
give  the  daily  rain- fall  in  inches.  This  method  of  representing  the  rain-fall 
will  do  for  this  chart,  but,  for  most  meteorological  purposes,  is  insufficient. 
The  time  of  the  commencement  and  end  of  the  rain-fall  should  be  given  where 
any  effect  due  to  the  rain  is  to  be  detected.  These  few  diagrams  illustrate  the 
method  of  graphical  representation,  so  that  any  one  should  with  little  difficulty 
be  able  now  to  make  them  for  such  cases  as  he  may  see  fit. 

On  pages  75,  76,  77,  are  some  designs,  showing  other  uses  to  which  squared 
or  quadrille  paper  can  be  put.  The  execution  of  such  ornamental  designs  is 
greatly  facilitated  by  the  use  of  this  paper.  The  figure  on  page  77  illustrates 
how  color  may  be  represented  in  a  design,  by  different  grades  and  directions, 
of  black  lines  and  white  spaces. 


DRAWING  INSTRUMENTS. 


NOVEMBER,  1873. 


FIG.  166. 


DRAWING  INSTRUMENTS. 


75 


76 


DRAWING   INSTRUMENTS. 


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DRAWING 


&*       «OR6 

#V 


ORTHOGRAPHIC  PROJECTION. 


ARCHITECTURAL  and  mechanical  drawings  are  usually  the  delineation  of 
bodies  by  orthographic  projection,  the  representation  on  a  sheet  of  paper  hav- 
ing only  two  dimensions,  length  and  breadth,  of  solids  having  three,  length, 
breadth,  and  thickness  ;  and  on  such  scales  that  dimensions  can  be  taken  from 
the  parts,  and  structures  and  machines  constructed  therefrom. 

Place  any  surface — for  instance,  a  sheet  of  paper  or  a  drawing-board — at 
right  angles  to  the  sun's  rays.  This  may  be  readily  done  by  inserting  a  pin  into 
the  surface,  and  making  it  vertical  to  the  surface  in  every  direction  by  a  right- 
angled  triangle  ;  then  place  the  surface  in  the  direct  rays  of  the  sun,  and  in 
such  a  position  that  there  will  be  no  shadow  on  the  surface  from  the  pin  ;  the 
sun's  rays  are  then  perpendicular  to  the  surface.  Take  a  wafer  or  a  circular 
bit  of  paper,  and  hold  it  over  the  paper  by  means  of  a  long  pin  or  wire,  and  we 
obtain  shadows,  as  above,  varying  with  the  inclination  of  the  wafer  to  the 
plane  of  the  paper.  When  parallel  with  the  plane,  the  shadow  is  a  complete 
circle  ;  when  at  right  angles,  a  line  ;  and  varying  between  them  as  the  wafer  is 
inclined.  These  shadows  are  the  orthographic  projections  of  the  wafer  ;  no 
line  can  be  longer  than  it  is  naturally,  but,  if  inclined  or  vertical,  it  is  reduced 
in  length  till  it  becomes  a  point  only.  The  orthographic  projection  of  the  pin 
which  has  determined  the  position  of  the  surface  is  merely  the  shadow  of  the 
head.  If  the  pin  be  inclined  at  all,  the  body  of  the  pin  is  projected  as  a  shadow 
by  a  line  ;  if  the  pin  be  laid  on  the  surface,  its  shadow,  or  projection,  is  that  of 
the  whole  length  of  the  pin.  The  sun's  rays  act  as  perpendiculars,  which 
will  be  hereafter  spoken  of  as  projecting  the  points  of  an  object  upon  a  surface 
which  will  represent  the  object  itself  in  drawing ;  and,  should  any  confusion 
occur  to  the  draughtsman  of  how  an  object  is' to  be  projected  or  drawn,  if  he 
can  make  the  outline  of  the  object  on  any  convenient  scale  in  wire  and  get  its 
shadows  by  the  sun's  vertical  rays  on  a  plane,  he  can  readily  see  how  the  object 
should  be  drawn. 

Since  the  surfaces  of  all  bodies  may  be  considered  as  composed  of  points, 
the  first  step  is  to  represent  the  position  in  space  of  a  point,  by  referring  it  to 
planes  whose  position  is  established.  The  projection  of  a  point  upon  a  plane 
is  the  foot  of  the  perpendicular  let  fall  from  the  point  on  the  plane.  If,  there- 


ORTHOGRAPHIC   PROJECTION. 


79 


iore,  on  two  planes  not  parallel  to  each  other,  whose  positions  are  known,  we  have 
the  projections  of  a  point,  the  position  of  this  point  is  completely  determined  by 
erecting  perpendiculars  from  each  plane  at  the  pro- 
jected points  :  their  intersection  will  be  the  point. 

If  from  every  point  of  an  indefinite  straight  line, 
A  B  (Fig.  167),  placed  in  any  manner  in  space,  per- 
pendiculars be  let  fall  on  a  plane,  L  M  N  0,  whose 
position  is  given,  then  all  the  points  in  which  these 
perpendiculars  meet  the  plane  will  form  another 
indefinite  straight  line,  a  b :  this  line  is  called  the 
projection  of  the  line  A  B  on  this  plane.  Since  two 
points  are  sufficient  to  determine  a  straight  line,  it 
is  only  necessary  to  project  two  points  of  the  line, 
and  the  straight  line  drawn  through  the  two  projected  points  will  be  the  pro- 
jection of  the  given  line.  The  projection  of  a  straight  line,  itself  perpendicular 
to  the  plane,  is  the  point  in  which  this  perpendicular  meets  the  plane. 

If  the  projections  a  1)  and  a'  V  of  a  straight  line  on  the  two  planes  L  M  N  0 
.and  L  M  P  Q  (Fig.  168)  are   known,  this  line   A  B   is   determined  ;  for  if, 


FIG.  167. 


FIG.  168. 


FIG.  169. 


through  one  of  its  projections,  a  #,  we  suppose  a  plane  drawn  perpendicularly 
to  L  M  N  0,  and  if  through  a'  V  another  plane  be  drawn  perpendicular  to 
L  M  P  Q,  the  intersection  of  the  two  planes  will  be  the  line  A  B. 

To  delineate  a  solid,  as  the  form  of  a  machine,  for  instance,  it  must  be 
referred  to  three  series  of  dimensions,  each  of  them  at  right  angles  to  the  plane 
of  the  other. 

Thus,  let  a  b  c  (Fig.  169)  be  a  parallelepiped  in  an  upright  position,  of 


80 


ORTHOGRAPHIC   PROJECTION. 


which  the  plane  a  b  is  horizontal,  and  the  planes  a  c  and  c  ~b  vertical.  Let  d  e, 
d  f,  and  d  g,  be  the  planes  of  projection.  The  sides  of  the  body  being  parallel 
to  these  planes,  each  to  each,  let  the  figure  of  the  parallelepiped  be  projected 
on  them.  For  this  purpose  draw  parallel  lines  from  the  angles  of  the  body 
perpendicular  to  the  planes,  as  indicated  by  the  dotted  lines  ;  then  upon  the 
plane  d  e  we  shall  have  a'  1)',  the  projection  of  the  surface  a  I :  this  is  called 
the  plan  of  the  object.  Upon  the  plane  dfwe  have  a'  c',  the  projection  of  the 
surface  a  c,  the  front  elevation  ;  and  upon  the  plane  d  g,  the  projection  I'  c' 
of  the  surface  b  c,  the  side  elevation.  We  have  then  three  distinct  views  of 
the  regular  solid  a  b  c  delineated  on  plane  surfaces,  which  convey  an  accurate 
and  sufficient  idea  of  its  form.  Indeed,  any  two  of  these  representations  are 
sufficient  as  a  description  of  the  object.  From  the  two  figures  a'  c',  5'  c',  for 
example,  the  third  figure  a'  b'  may  be  compounded,  by  merely  drawing  the 
vertical  lines  c'  h  b'  i,  and  a'  k,  c'  I,  to  meet  the  plane  d  e,  and  by  producing 
them  horizontally  till  they  meet  and  form  the  figure  a'  b'.  Similarly,  the 
figure  b'  c'  may  be  deduced  from  the  other  two  by  the  aid  of  the  lines  Ji,  i, 
from  a'  b1 ',  and  the  lines  m,  n,  from  a'  c' . 

It  is  in  this  way  that  a  third  view  of  any  piece  of  machinery  is  to  be  found 
from  two  given  views  ;  and  in  many  cases  two  elevations,  or  one  elevation  and 

a  plan,  may  afford  a  sufficiently  corn- 
plete idea  of  the  construction  of  a 
machine.  In  other  cases,  many  parts 
may  be  concealed  by  others  in  which 
they  are  inclosed  ;  this  suggests  the 
occasional  necessity  of  views  of  the 
interior,  in  which  the  machine  is  sup- 
posed to  be  cut  across  by  planes,  ver- 
tically or  horizontally,  so  as  properly 
to  reveal  its  structure.  Such  views 
are  termed  sections,  and,  with  refer- 
ence to  the  planes  of  section,  are  de- 
nominated vertical  and  horizontal  sec- 
tions. To  all  such  drawings  is  given 
the  general  title  of  geometrical  draw- 
ings, as  distinguished  from  perspective 
drawings. 

In  practice,  the  drawings  are  done 
upon  one  common  surface,  the  plane 
of  paper,  and  we  may  readily  suppose 
the  plane  d  g  (Fig.  169)  revolved  back 
into  the  position  d  gr,  and  d  e  also 
moved  to  d  e',  both  of  these  positions 

being  in  the  plane  of  d  f.  This  being  done,  we  have  the  three  views  depicted  on 
one  plane  surface  (Fig.  170).  In  this  figure,  the  same  letters  of  reference  are 
employed  as  in  Fig.  169  ;  d  I  and  d  m  are  the  ground  and  vertical  lines.  It  is 
evident  that  the  positions  of  the  same  points  in  a'  c'  and  a'  b'  are  in  the  same 
perpendicular  from  the  ground-line  :  that,  in  short,  the  position  of  a  point  in 


FIG.  170. 


ORTHOGRAPHIC  PROJECTION.  81 

the  plane  may  be  found  by  applying  the  edge  of  the  square  to  the  same  point 
as  represented  in  the  elevation.  The  same  remark  is  applicable  as  between  the 
two  elevations.  Hence  the  method  of  drawing  several  views  of  one  machine 
upon  the  same  surface  of  paper  in  strict  agreement  with  each  other. 

PROJECTIONS   OF   SIMPLE   BODIES. 

In  most  of  the  following  examples,  the  projections  of  the  bodies  are  given 
both  with  and  without  the  construction  lines. 

Right  projections  of  a  regular  hexagonal  pyramid  (Fig.  171). — It  is  evident 
that  two  distinct  geometrical  views  are  necessary  to  convey  a  complete  idea  of 
the  form  of  the  object  :  an  elevation  to  represent  the  sides  of  the  body,  and  to 
express  its  height  ;  and  a  plan  to  express  the  form  horizontally. 

Draw  a  horizontal  straight  line  L  T  through  the  center  of  the  sheet  to  rep- 
resent the  ground-line.  Then  draw  a  perpendicular  S  S'  to  the  ground-line  to 
represent  the  axis  of  the  pyramid.  For  the  sake  of  preserving  the  symmetry  of 
the  drawing,  the  centers  of  the  horizontal  projections  of  Figs.  171  and  172  are 
in  the  same  straight  line  A'  S',  drawn  parallel  to  the  ground-line. 

In  delineating  the  pyramid,  it  is  necessary,  in  the  first  place,  to  construct  the 
plan.  Take  any  point,  S',  on  the  line  S  S'  as  the  center  of  the  figure,  and  from 
this  point,  with  a  radius  equal  to. the  side  of  the  hexagon  which  forms  the  base 
of  the  pyramid,  describe  a  circle,  cutting  A'  S'  at  A'  and  D'.  From  these  points 
with  the  same  radius,  draw  four  arcs  of  circles,  cutting  the  primary  circle  in 
four  points.  These  six  points  being  joined  by  straight  lines,  will  form  the  figure 
A'  B'  0'  D'  E'  F',  the  base  of  the  pyramid ;  and  the  lines  A'  S',  B'  S',  etc.,  will 
represent  the  projections  of  its  edges  shortened  as  they  would  appear  in  the  plan. 

By  the  help  of  this  plan  the  vertical  projection  of  the  pyramid  may  be  easily 
constructed.  Since  its  base  rests  upon  the  horizontal  plane,  it  must  be  pro- 
jected vertically  upon  the  ground-line  ;  therefore,  from  each  of  the  angles  at 
A',  B',  C',  and  D',  erect  perpendiculars  to  that  line.  The  points  of  intersec- 
tion, A,  B,  C,  and  D,  are  the  true  positions  of  all  the  angles  of  the  base  ;  and 
it  only  remains  to  lay  off  the  height  of  the  pyramid,  from  the  point  G  to  S,  and 
to  draw  S  A,  S  B,  S  C,  and  S  D,  which  are  the  only  edges  of  the  pyramid  visi- 
ble in  the  elevation.  Of  these  it  is  to  be  remarked  that  S  A  and  S  D  alone, 
being  parallel  to  the  vertical  plane,  are  seen  in  their  true  length  ;  and,  more- 
over, that  from  the  assumed  position  of  the  solid  under  examination,  the  points 
F'  and  E'  being  situated  in  the  lines  B  B'  and  C  C',  the  lines  S  B  and  S  C  are 
each  the  projections  of  two  edges  of  the  pyramid. 

To  construct  the  projections  of  the  same  pyramid,  having  its  base  set  in  an 
inclined  position,  but  with  its  edges  S  A  and  S  D  still  parallel  to  the  vertical 
plane  (Fig.  172). 

It  is  evident  that,  with  the  exception  of  the  inclination,  the  vertical  projec- 
tion of  this  solid  is  precisely  the  same  as  in  the  preceding  example,  and  it  is 
only  necessary  to  copy  that  elevation.  To  do  this,  fix  the  position  of  the  point  D 
upon  the  ground-line,  through  which  draw  D  A,  making  with  L  T  the  desired 
inclination  of  the  base  of  the  pyramid.  Make  D  A  equal  to  the  A  D  of  the 
preceding  figure,  and  on  this  erect  the  vertical  projection  S  A  D  of  that  figure. 

Since  the  edges  S  A  and  S  D  are  still  parallel  to  the  vertical  plane,  and 
6 


82 


ORTHOGRAPHIC   PROJECTION. 


FIG.  171. 


FIG.  172. 


ORTHOGRAPHIC   PROJECTION. 


83 


the  point  D  remains  unaltered,  the  projection  A'  of  the  point  A  will  still  be  in 
the  line  M  N.  The  remaining  points  B',  C',  etc.,  in  the  projection  of  the  base, 
are  found  by  the  intersections  of  perpendiculars  let  fall  from  the  corresponding 
points  in  the  elevation,  with  lines  drawn  parallel  to  M  N,  at  a  distance  equal 
to  the  width  of  the  base.  By  joining  all  the  contiguous  points,  we  obtain 
A'  B'  C'  D'  E'  F',  the  horizontal  projection  of  the  base,  two  of  its  sides,,  how- 
ever, are  dotted,  being  concealed  by  the  body  of  the  pyramid.  The  vertex  S 
having  been  similarly  projected  to  S',  and  joined  by  straight  lines  to  the  several 
angles  of  the  base,  the  projection  of  the  solid  is  completed. 

To  find  the  horizontal  projection  of  a  transverse  section  of  the  same  pyramid, 
made  by  a  plane  perpendicular  to  the  vertical,  but  inclined  at  an  angle  to  the 
horizontal  plane  of  projection;  and  let  all  the  sides  of  the  base  be  inclined  to  the 
ground-line  (Fig.  173). 


FIG.  173. 

Since  none  of  the  sides  of  the  base  are  to  be  parallel  with  the  ground-line, 
draw  a  diameter  A'  D'  making  the  required  angle  with  that  line,  and  from  the 
points  A'  and  D'  proceed  to  set  out  the  angular  points  of  the  hexagon  as  in  the 
figure.  Then,  in  order  to  obtain  the  projections  of  the  edges  of  the  pyramid, 
join  the  angular  points  which  are  diametrically  opposite  ;  and,  following  the 


ORTHOGRAPHIC   PROJECTION. 


method  pointed  out  in  reference  to  Fig.  171,  project  the  figure  thus  obtained 
upon  the  vertical  plane,  as  shown  in  the  elevation. 

Now,  if  the  cutting  plane  be  represented  by  the  line  a  d  in  the  elevation,  it 
is  obvious  that  it  will  expose,  as  the  section  of  the  pyramid,  a  polygon  whose 
angular  points,  being  the  intersections  of  the  various  edges  with  the  cutting 
plane,  will  be  projected  in  perpendiculars  drawn  from  the  points  where  it 
meets  these  edges  respectively.  If,  therefore,  from  the  points  a,  f,  b,  etc.,  we 
let  fall  the  perpendiculars  a  a',  //',  b  bf,  etc.,  and  join  their  contiguous  points 
of  intersection  with  the  lines  A'  D',  F'  C',  B'  E',  etc.,  we  shall  form  a  six-sided 
figure,  which  will  represent  the  section  required.  The  edges  F  S  and  E  S,  being 
concealed  in  the  elevation,  but  necessary  for  the  construction  of  the  plan,  have 
been  expressed  in  dotted  lines,  as  also  the  portion  of  the  pyramid  situated  above 
the  cutting  plane,  which,  though  supposed  to  be  removed,  is  necessary  in  order 
to  draw  the  lines  representing  the  edges.  We  have  here  introduced  the  ordi- 
nary method  of  expressing  sections  in  purely  line-drawings,  by  filling  up  the 
spaces  comprised  within  their  outlines  with  a  number  of  parallel  straight  lines 
drawn  at  equal  distances  called  section-lines. 

PROJECTIONS   OF  A   PEISM. 


B 


a       H 


C 


D 


I  K 


FIG.  174. 


ORTHOGRAPHIC   PROJECTION. 


85 


Required  to  represent  in  plan  and  elevation  a  regular  six-sided  prism  in  an 
upright  position  (Fig.  174). 

Lay  down  the  ground-line  G  K  and  draw  the  axis  of  the  prism  S  S'.  De- 
scribe the  hexagonal  plan  A'  B'  C'  D'  E'  F',  as  in  the  previous  example.  From 
each  of  the  angular  points,  A',  B',  etc.,  erect  perpendiculars  to  the  ground-line, 
and  on  one  of  these  perpendiculars  set  off  A  G,  the  height  of  the  prism,  and 
draw  a  parallel  A  D  to  the  ground-line,  which  completes  the  vertical  projection. 
The  face,  B  0  H  I,  being  parallel  to  the  vertical  plane,  is  seen  in  its  true  size. 
B'  C'  being  equal  to  one  half  of  A'  D',  therefore  H  I  is  equal  to  one  half  of 
G  K.  We  have  then  G  H  and  I  K  equal  each  to  one  half  of  H  I.  This  enables 
us  to  draw  the  elevation  of  such  a  prism  situated  as  is  this  one  without  con- 
structing the  plan.  This  fact  should  be  remembered  in  the  drawing  of  nuts, 
bolt-heads,  etc.,  in  machine-drawing,  where  it  is  of  frequent  application. 

To  form  the  projections  of  the  same  prism,  supposing  it  to  have  been  moved 
round  the  point  G,  in  a  plane  parallel  to  the  vertical  plane  (Fig.  175). 

Copy  the  elevation  (Fig.  174)  on  the  inclined  base  G  K.     Let  fall  perpen- 


FIG.  175. 


86 


ORTHOGRAPHIC  PROJECTION. 


diculars  from  all  the  angles  in  the  elevation,  and,  joining  the  contiguous  points 
of  intersection  with  the  horizontal  lines  appropriate  to  these  points  respectively, 
the  plan  of  course  remaining  the  same  width  as  before,  we  obtain  the  polygon 
A'  B'  0'  D'  E'  F'  as  the  projection  of  the  upper  surface,  and  G'  H'  I'  K'  L'  M' 
as  that  of  the  base  of  the  prism.  Finally,  it  will  be  observed  that  all  the  edges 
are  represented,  in  the  horizontal  projection,  by  equal  straight  lines,  as  D'  K', 
A'  G',  etc.,  and  that  the  sides  A'  B',  G'  H',  etc.,  remain  still  parallel  to  each 
other,  which  will  afford  the  means  of  verifying  the  accuracy  of  the  drawings. 
As  the  upper  surface  and  the  base  are  seen  obliquely  in  this  projection,  of 
course  they  do  not  appear  as  true  hexagons  in  the  plan. 

Required  the  projections  of  the  same  prism  set  into  a  position  inclined  to 
loth  planes  of  projection  (Fig.  176). 


FIG.  176. 


Assuming  that  the  inclination  of  the  prism  upon  the  horizontal  plane  is 
the  same  as  in  the  preceding  figures,  for  the  sake  of  simplifying  the  operation, 
the  first  process  is  to  copy  the  plan  of  Fig.  175  on  an  axis  A'  K'  inclined  to  the 
vertical  plane  of  projection. 


ORTHOGRAPHIC   PROJECTION. 


87 


Now,  since  the  prism  has  been  supposed  to  have  preserved  its  former  inclina- 
tion to  the  horizontal  plane,  it  is  obvious  that  every  point  in  it,  such  as  A,  has, 
in  assuming  its  new  position,  simply  moved  in  a  horizontal  plane,  and  will 
therefore  be  at  the  same  distance  above  the  ground-line  that  it  was  in  the 
elevation  (Fig.  175),  and  it  will  also  be  in  the  perpendicular  A' A  ;  the  point 
of  intersection  A  is,  therefore,  its  projection  in  the  elevation.  The  remaining 
angular  points  in  this  view  are  all  determined  in  the  same  manner,  and,  having 
joined  the  contiguous  points,  and  the  corresponding  angles  of  the  upper  and 
lower  surface,  we  obtain  the  complete  vertical  projection  of  the  prism  in  its 
doubly-inclined  position. 


CONSTKUCTION   OF  THE   CONIC   SECTIONS. 

The  plan  of  the  cone  (Fig.  177)  is  simply  a  circle,  described  from  the  center 
S',  with  a  diameter  equal  to  that  of  the  base.  Its  elevation  is  an  isosceles  tri- 
angle, obtained  by  drawing  tangents  A'  A,  B'  B,  perpendicular  to  and  inter- 


X 


X 


FIG.  177. 


88  ORTHOGRAPHIC   PROJECTION. 

secting  the  ground-line ;  then  set  off  upon  the  center  line  the  height  C  S,  and 
join  S  A,  S  B.  These  lines  are  called  the  exterior  elements  of  the  cone. 

Given  the  projections  of  a  cone,  and  the  direction  of  a  plane  X  X,  cutting  it 
perpendicularly  to  the  vertical,  and  obliquely  to  the  horizontal  plane  ;  required 
to  find,  first,  the  horizontal  projection  of  this  section;  and,  secondly,  the  out- 
line of  the  ellipse  thus  formed  (Figs.  177,  178). 

Through  the  vertex  of  the  cone  draw  a  line  S  E  to  any  point  within  the 
base  A  B  ;  let  fall  a  perpendicular  from  E,  cutting  the  circumference  of  the  base 
in  E',  and  join  E'  S' ;  then  another  perpendicular  let  fall  from  e  will  intersect 
E'  S'  in  a  point  ef,  which  will  be  the  horizontal  projection  of  a  point  in  the 
curve  required  ;  and  so  on  for  any  required  number  of  points. 

The  exterior  generatrices  A  S  and  B  S  being  both  projected  upon  the  line 
A'  B',  the  extreme  limits  of  the  curve  sought  will  be  at  the  points  a'  and  bf  on 
that  line,  which  are  the  projections  of  the  points  of  intersection  a  and  b  of  the 
cutting  plane  with  the  outlines  of  the  cone.  And  since  the  line  a'  b'  will 
obviously  divide  the  curve  symmetrically  into  two  equal  parts,  the  points  /', 
g ' ,  h' ,  etc.,  will  be  readily  obtained  by  setting  off  above  that  line,  and  on  their 
respective  perpendiculars,  the  distances  d'  d*,  e'  e*,  etc.  A  sufficient  number 
of  points  having  thus  been  determined,  the  curve  drawn  through  them  (which 
will  be  found  to  be  an  ellipse)  will  be  the  outline  of  the  section  required. 

This  curve  may  be  obtained  by  another  method,  depending  on  the  principle 
that  all  sections  of  a  cone  by  planes  parallel  to  the  base  are  circles.  Thus,  let 
the  line  F  G  represent  such  a  cutting  plane  ;  the  section  which  it  makes  with 
the  cone  will  be  denoted  on  the  horizontal  projection  by  a  circle  drawn  from 
the  center  S',  with  a  radius  equal  to  half  the  line  F  G  ;  and  by  projecting  the 
point  of  intersection  H  of  the  horizontal  and  oblique  planes  by  a  perpendicular 
H  H',  and  noting  where  this  line  cuts  the  circle  above  referred  to,  we  obtain 
two  points  H'  and  I'  in  the  curve  required.  By  a  similar  construction,  as 
exemplified  in  the  drawings,  any  number  of  additional  points  may  be  found. 

As  the  projection  obtained  by  the  preceding  methods  exhibits  the  section  as 
fore-shortened,  and  not  in  its  true  dimensions,  we  shall  now  proceed  to  the 
consideration  of  the  second  question  proposed.  Let  the  cutting  plane  X  X  be 
conceived  to  turn  upon  the  point  b,  so  as  to  coincide  with  the  vertical  line  b  k, 
and  (to  avoid  confusion  of  lines)  let  b  Tc  be  transferred  to  a'  b',  which  will  rep- 
resent, as  before,  the  extreme  limits  of  the  curve  required.  Now,  taking  any 
point,  such  as  d,  it  is  obvious  that,  in  this  new  position  of  the  cutting  plane,  it 
will  be  represented  by  d?,  and,  if  the  cutting  plane  were  turned  upon  a'  b'  (Fig. 
178)  as  an  axis  till  it  is  parallel  to  the  vertical  plane,  the  point  which  had  been 
projected  at  d*  would  then  have  described  round  a'  b'  an  arc  of  a  circle,  whose 
radius  is  the  distance  d'  d?  (Fig.  177).  This  distance,  therefore,  being  set  off 
at  d'  and  f  on  each  side  of  a'  b' ,  gives  two  points  in  the  curve  sought.  By 
a  similar  mode  of  operation  any  number  of  points  may  be  obtained,  through 
which,  if  a  curve  be  drawn,  it  will  be  an  ellipse  of  the  true  form  and  dimen- 
sions of  the  section. 

To  find  the  horizontal  projection  and  actual  outline  of  the  section  of  a  cone, 
made  by  a  plane  Y  Y  parallel  to  one  side  or  element,  and  perpendicular  to  the 
vertical  plane  (Figs.  179,  180). 


ORTHOGRAPHIC   PROJ 


89 


Determine  by  the  second  method  laid  down  in  the  preceding  problem  any 
number  of  points,  as  F',  G',  J',  K',  etc.,  in  the  curve  representing  the  horizon- 
tal projection  of  the  section  specified.  »  The  horizontal  plane  passing  through 
M  gives  only  one  point  M',  which  is  the  vertex  of  the  curve  sought. 


FIG.  180. 

In  order  to  determine  the 
actual  outline  of  this  curve, 
suppose  the  plane  Y  Y  to  turn 
as  upon  a  pivot  at  M,  until  it 
has  assumed  the  position  M  B, 
and  transfer  M  B  parallel  to 
itself  to  M2  B2  (Fig.  180).  The 
point  F  will  thus  have  first 
described  the  arc  F  E  till  it 
reaches  the  point  E,  which  is 
then  projected  to  E2 ;  suppose 
the  given  plane,  now  represent- 
ed by  M2  B2,  to  turn  upon  that 
line  as  an  axis,  until  it  assumes 
a  position  parallel  to  the  ver- 
tical plane,  the  point  E2,  which  is  distant  from  the  axis  M'  B'  by  the  distance 
F'  S'  (Fig.  179),  will  now  be  projected  to  F2  and  Ga,  two  points  in  the  curve 
required,  which  is  &  parabola. 

To  draw  the  vertical  projection  of  the  sections  of  two  opposite  cones  made  ly 
a  plane  parallel  to  their  axis  (Fig.  181). 

Let  C  E  D  and  C  B  A  be  the  two  cones,  and  X  X  the  position  of  the 
cutting  plane.  Project  in  plan  either  of  the  cones,  as  I  E'  D'  ;  from  its  center, 
with  a  radius  equal  to  L  H,  describe  a  circle,  and  draw  the  tangent  la;  la 
will  be  the  horizontal  projection  of  the  cutting  plane.  Draw  the  line  H'  M' 
parallel  to  the  cutting  plane  ;  H',  M'  corresponding  in  position  to  the  inter- 


90 


ORTHOGRAPHIC   PROJECTION. 


sections  H,  M,  of  the  plane  with  the  cones.  From  H'  and  M'  lay  off  distances 
equal  to  L  K,  K  I,  and  the  length  of  the  cone,  and  through  these  points  draw 
perpendiculars,  as  f  e\  d'  cr ,  V  a',  etc.,  which  must  be  made  equal  to  the 
chords  f  e,  d  c,  b  a,  made  by  the  cutting  plane  a  b,  with  circles  whose  radii  are 


G  K,  -I  F,  and  the  radius  of  the  base  of  the  cone.  Through  the  points  a',  c', 
e,  H',  /',  d',V,  draw  the  curve,  and  we  have  the  projection  required.  A  similar 
construction  will  give  the  sectional  projection  of  the  opposite  cone  at  M'.  The 
curve  thus  found  is  the  hyperbola. 


PENETRATIONS   OR   INTERSECTIONS   OF   SOLIDS. 

On  examining  the  minor  details  of  most  machines,  we  find  numerous  ex- 
amples of  cylindrical  and  other  forms,  fitted  to,  and  even  appearing  to  pass 
through,  each  other  in  a  great  variety  of  ways.  The  examples  given  are  selected 
with  a  view  of  exhibiting  those  cases  which  are  of  most  frequent  occurrence, 
and  of  elucidating  general  principles. 

Represent  the  projections  of  two  cylinders  of  unequal  diameters  (Fig.  182) 
meeting  each  other  at  right  angles  ;  one  by  the  rectangle  ABED  in  the  ver- 
tical, and  by  the  circle  A'  H'  B'  in  the  horizontal  projections  ;  the  other,  which 
is  supposed  to  be' horizontal,  is  indicated  in  the  former  by  the  circle  L  P  I  N, 
and  in  the  latter  by  the  rectangle  L'  I'  K'  M'.  From  the  position  of  these  two 
solids  it  is  evident  that  the  curves  formed  by  their  junction  will  be  projected 
horizontally  in  the  curves  0'  H'  P',  R'  S'  T',  and  vertically  by  L  P  I  N. 

But,  if  the  position  of  these  bodies  be  changed  into  that  represented  by  Fig. 
183,  the  lines  of  their  intersection  will  assume  in  the  vertical  projection  a 
totally  different  aspect,  and  may  be  accurately  determined  as  follows  : 

Through  any  point  taken  upon  the  plan  of  Fig.  183  draw  a  horizontal  line 
a'  V,  which  is  to  be  considered  as  indicating  a  plane  cutting  both  cylinders 
parallel  to  their  axes  ;  this  plane  would  cut  the  vertical  cylinder  in  lines  drawn 
perpendicularly  through  the  points  c'  and  d'.  To  find  the  vertical  projection 
of  its  intersection  with  the  other  cylinder,  conceive  its  base  I'  L',  after  being 


ORTHOGRAPHIC   PROJECTION. 


91 


FIG.  182. 


FIG.  183. 


92  ORTHOGRAPHIC  PROJECTION. 

transferred  to  I2  La,  to  be  revolved  about  I2  L2  as  an  axis  parallel  to  the  hori- 
zontal plane  ;  this  is  expressed  in  part  by  simply  drawing  a  semicircle  of  the 
diameter  I3  L1.  Produce  the  line  a'  V  to  #a  ;  then  set  off  the  distance  a?  e'  on 
each  side  of  the  axis  I  K,  and  draw  straight  lines  through  these  points  parallel 
to  it.  These  lines  a  b,  g  h,  denote  the  intersection  of  the  plane  a'  V  with  the 
horizontal  cylinder,  and  therefore  the  points  c,  d,  m,  o,  where  they  cut  the 
perpendiculars  c  c',  d  d',  are  points  in  the  curve  required.  By  passing  other 
horizontal  planes  similar  to  a'  V  through  both  cylinders,  and  operating  as 
before,  any  number  of  points  may  be  obtained.  The  vertices  i  and  k  of  the 
curves  are  obviously  projected  directly  from  i'  and  Jc',  the  intersections  of  the 
outlines  of  both  cylinders.  When  the  cylinders  are  of  unequal  diameters,  as 
in  the  present  case,  the  curves  of  penetration  are  hyperbolas. 

When  the  diameters  of  the  cylinders  are  equal  (Fig.  184),  and  when  they 
cut  each  other  at  right  angles,  the  curves  of  penetration  are  projected  vertically 
in  straight  lines  perpendicular  to  each  other.  In  the  figure,  most  of  the  points 
are  indicated  in  elevation  and  plan  by  the  same  letters  of  reference. 

To  delineate  the  intersections  of  two  cylinders  of  equal  diameters  at  right 
angles,  when  one  of  the  cylinders  is  inclined  to  the  vertical  plane  (Fig.  185). 

Supposing  the  two  preceding  figures  to  be  drawn,  the  projection  c  of  any 
point  such  as  c'  may  be  ascertained  by  observing  that  it  must  be  situated  in 
the  perpendicular  c'  c,  and  that,  since  the  distance  of  this  point  (projected  at  c 
in  Fig.  184)  from  the  horizontal  plane  remains  unaltered,  it  must  also  be  in 
the  horizontal  line  c  c.  Upon  these  principles  all  the  points  indicated  by  literal 
references  in  Fig.  185  are  determined  ;  the  curves  of  penetration  resulting 
therefrom  intersecting  each  other  at  two  points  projected  upon  the  axial  line 
L  K,  of  which  that  marked  q  alone  is  seen.  The  ends  of  the  horizontal  cylin- 
der are  represented  by  ellipses,  the  construction  of  which  will  also  be  obvious 
on  referring  to  the  figure. 

To  find  the  curves  resulting  from  the  intersection  of  two  cylinders  of  un- 
equal diameters,  meeting  at  any  angle  (Fig.  186). 

For  the  sake  of  simplicity,  suppose  the  axes  of  both  cylinders  to  be  parallel 
to  the  vertical  plane,  and  let  A  B  E  D  and  N  0  Q  P  be  their  projections  upon 
that  plane.  In  constructing,  in  the  first  place,  their  horizontal  projection, 
observe  that  the  upper  end  A  B  of  the  larger  cylinder  is  represented  by  an 
ellipse  A'  K'  B'  M',  which  may  easily  be  drawn  by  the  help  of  the  major  axis 
K'  M'  equal  to  the  diameter  of  the  cylinder,  and  of  the  minor  A'  B',  the  projec- 
tion of  the  diameter.  The  visible  portion  of  the  base  of  the  cylinder  being 
similarly  represented  by  the  semi-ellipse  L'  D'  II',  its  entire  outline  will  be  com- 
pleted by  drawing  tangents  L'  M'  and  H'  K'.  The  upper  extremity  P  N  of  the 
smaller  cylinder  will  also  be  projected  in  the  ellipse  p'  i'  N'. 

Conceive  a  plane,  as  a'  g',  to  pass  through  both  cylinders  parallel  to  their 
axes  ;  it  will  cut  the  surface  of  the  larger  cylinder  in  two  straight  lines,  passing 
through  the  points/'  and  g'  on  the  upper  end  of  the  cylinder  ;  these  lines  will 
be  represented  in  the  elevation,  by  projecting  the  points/'  and  g'  to/,  g  ;  and 
drawing  a  f  and  c  g  parallel  to  the  axis.  The  plane  a'  g'  will  in  like  manner 
cut  the  smaller  cylinder  in  two  straight  lines,  which  will  be  represented  in  the 
vertical  projection  by  d  h  and  e  i,  and  the  intersections  of  these  lines  with  af 


ORTHOGRAPHIC  PROJECTION. 


93 


FIG.  184. 


FIG.  185. 


ORTHOGRAPHIC   PROJECTION". 


and  c  g  will  give  four  points  ?,  &,  m,  and  n,  in  the  curves  of  penetration.  Of 
these  points,  one  only,  that  marked  I,  is  visible  in  the  plan,  where  it  is  denoted 
by/'. 

To  find  the  curves  of  penetration  in  the  elevation  without  the  aid  of  the  plan 
(Fig.  186). 

Let  the  bases  D  E  and  Q  0  of  both  cylinders  be  conceived  to  be  revolved 
parallel  to  the  vertical  plane  after  being  transferred  to  any  convenient  distance, 

as  D2  E2  and  Q2  O2,  from  the 
principal  figure  ;  they  will 
then  be  vertically  projected 
in  the  circles  D2  H2  E2  and 
Q2  G'  0".  Now  draw  «2  e* 
parallel  to  D  E,  and  at  any 
suitable  distance  from  the 
center  I  ;  this  line  will  rep- 
resent the  intersection  of  the 
base  of  the  cylinder  with  a 
plane  parallel  to  the  axes  of 
both,  as  before.  The  inter- 
section of  this  plane  with  the 
base  of  the  smaller  cylinder 
will  be  found  by  setting  off 
from  R  a  distance  R  p,  equal 
to  I  o,  and  drawing  through 
(i^G  .  \0"  the  point  p  a  straight  line 
G[_  \ JR, .--•' '\  '|  parallel  to  Q  0.  It  is  obvious 
that  the  intersection  of  the 
supposed  plane  with  the  con- 
vex surfaces  of  the  cylinders 
will  be  represented  by  the 
lines  a  f.,  c  g,  and  d  h,  e  i ; 
and  that,  consequently,  the 
intersections  of  these  lines 
indicate  points  in  the  curves 
sought.  These  points  may 
be  multiplied  indefinitely  by 
conceiving  other  planes  to 
pass  through  the  cylinders, 
and  operating  as  before. 
To  find  the  curves  of  penetration  of  a  cone  and  sphere  (Fig.  187). 
Let  D  S  be  the  axis  of  the  cone,  A'  L'  B'  the  circle  of  its  base,  and  the  tri- 
angle'A  B  S  its  projection  on  the  vertical  plane  ;  and  let  C,  C',  be  the  projec- 
tions of  the  center,  and  the  equal  circles  E'  K'  F'  and  E  G  F  those  of  the 
circumferences  of  the  sphere. 

This  problem,  like  most  others  similar  to  it,  can  be  solved  only  by  the  aid  of 
imaginary  intersecting  planes.  Let  a  b  represent  the  projection  of  a  horizontal 
plane  ;  it  will  cut  the  sphere  in  a  circle  whose  diameter  is  a  ~b,  and  which  is  par- 


FIG.  186. 


K 


ORTHOGRAPHIC   PROJECTION. 


95 


tially  drawn  from  the  center  C'  in  the  plan,  as  a'  f  V.  Its  intersection  with 
the  cone  is  also  a  circle  described  from  the  center  S'  with  the  diameter  c  d  as 
c'f  d' ;  the  points  e'  and/',  where  these  two  circles  cut  each  other,  are  the  hori- 
zontal projections  of  two  points  in  the  lower  curve,  which  is  evidently  entirely 
hidden  by  the  sphere.  The  points  referred  to  are  projected  vertically  upon  the 
line  a  b  at  e  and/.  The  upper  curve,  which  is  seen  in  both  projections,  is  ob- 


FIG.  187. 

tained  by  a  similar  process  ;  but  it  is  to  be  observed  that  the  horizontal  cutting 
planes  must  be  taken  in  such  positions  as  to  pass  through  both  solids  in  circles 
which  shall  intersect  each  other.  For  our  guidance  in  this  respect  it  will 
be  necessary,  first,  to  determine  the  vertices  m  and  n  of  the  curves  of  pene- 
tration. 

For  this  purpose,  conceive  a  vertical  plane  passing  through  the  axis  of  the 
cone  and  the  center  of  the  sphere  ;  its  horizontal  projection  will  be  the  straight 
line  C'  L'  joining  the  centers  of  the  two  bodies.  Let  us  also  make  the  suppo- 
sition that  this  plane  is  turned  upon  the  line  C  C'  as  on  an  axis,  until  it  be- 


96 


ORTHOGRAPHIC  PROJECTION. 


comes  parallel  to  the  vertical  plane  ;  the  points  S'  and  L'  will  now  have  assumed 
the  positions  S2  and  La,  and  consequently  the  axis  of  the  cone  will  be  projected 

vertically  in  the  line  D'  S3, 
and  its  side  in  S3  L3,  cut- 
ting the  sphere  at  the 
points  p  and.  r.  Conceive 
the  solids  to  have  resumed 
their  original  relative  po- 
sitions, it  is  clear  that  the 
vertices  or  adjacent  lim- 
iting points  of  the  curves 
of  penetration  must  be  in 
the  horizontal  lines  p  o 
and  r  q,  drawn  through 
the  points  determined  as 
above  ;  their  exact  posi- 
tions on  these  lines  may 
be  ascertained  by  project- 
ing vertically  the  points 
m'  and  ri,  where  the  arcs 
described  by  the  points  p 
and  r,  in  restoring  the 
cone  to  its  first  position, 
intersect  the  line  S'  L'. 

It  is  of  importance, 
further,  to  ascertain  the 
points  at  which  the  curves 
of  penetration  meet  the 
outlines  A  S  and  S  B  of  the 
cone.  The  plane  which 
passes  through  these  lines, 
being  projected  horizon- 
tally in  A'  B',  will  cut  the 
sphere  in  a  circle  whose 
diameter  is  i'  f  ;  this  cir- 
cle, described  in  the  ele- 
vation from  the  center  C, 
will  cut  the  sides  A  S  and 
S  B  in  four  points,  at 
which  the  curves  of  pene- 
tration are  tangent  to  the 
outlines  of  the  cone. 

To  find   the  lines  of 
penetration  of  a  cylinder 
and  a  cylindrical  ring  or 
torus  (Fig.  188). 
Let  the  circles  A'  E'  B',  F'  G'  K',  represent  the  horizontal,  and  the  figure 


ORTHOGKAPHIC   PROJECTION.  97 

A  C  B  D  the  vertical  projection  of  the  torus,  and  let  the  circle  H'/'  L',  and 
the  rectangle  II  I  M  L  be  the  analogous  projections  of  the  cylinder,  which 
passes  perpendicularly  through  it.  Conceive,  as  before,  a  plane,  a  b,  to  pass 
horizontally  through  both  solids  ;  it  will  obviously  cut  the  cylinder  in  a  circle 
which  will  be  projected  in  the  base  H'  /'  L'  itself,  and  the  ring  in  two  other 
circles,  of  which  one  only,  part  of  which  is  represented  by  the  arc/'  b*  b',  will 
intersect  the  cylinder  at  the  points/'  and  b3,  which,  being  projected  vertically, 
will  give  two  points /and  V*  in  the  upper  curve  of  penetration. 

Another  horizontal  plane,  taken  at  the  same  distance  below  the  center  line 
A  B  as  that  marked  a  b  is  above  it,  will  evidently  cut  the  ring  in  circles  coin- 
ciding with  those  already  obtained  ;  consequently  the  points/'  and  b9  indicate 
points  in  the  lower  as  well  as  in  the  upper  curves  of  penetration,  and  are  pro- 
jected vertically  at  d  and  e.  Thus,  by  laying  down  two  planes  at  equal  dis- 
tances on  each  side  of  A  B,  by  one  operation  four  points  in  the  curves  required 
are  determined. 

To  determine  the  vertices  m  and  n,  following  the  method  explained  in  the 
preceding  problem,  draw  a  plane  0  n',  passing  through  the  axis  of  the  cylinder 
and  the  center  of  the  ring,  and  conceive  this  plane  to  be  revolved  about  the 
point  0  until  it  has  assumed  the  position  0  B',  parallel  to  the  vertical  plane  ; 
the  point  n',  representing  the  extreme  outline  of  the  cylinder  in  plan,  will  now 
be  at  r',  and,  being  projected  vertically,  that  outline  will  cut  the  ring  in  two 
points  jo  and  r,  which  would  be  the  limits  of  the  curves  of  penetration  in  the 
supposed  relative  position  of  the  two  solids  ;  and  by  drawing  the  two  horizontal 
lines  r  n  and  p  m,  and  projecting  the  point  nr  vertically,  the  intersections  of 
these  lines,  m  and  n,  are  the  vertices  of  the  curves  in  the  actual  position  of  the 
penetrating  bodies. 

The  points  at  which  the  curves  are  tangents  to  the  outlines  H  I  and  L  M  of 
the  cylinder,  may  readily  be  found  by  describing  arcs  of  circles  from  the  center 
0  through  the  points  H'  and  L',  which  represent  these  lines  in  the  plan,  and 
then  proceeding,  as  above,  to  project  the  points  thus  obtained  upon  the  eleva- 
tion. Lastly,  to  determine  the  points,  as  j,  z,  etc.,  where  the  curves  are  tan- 
gents to  the  horizontal  outlines  of  the  ring,  draw  a  circle  P'  s'  j'  with  a  radius 
equal  to  that  of  the  center  line  of  the  ring,  namely,  P.  D  ;  the  points  of  inter- 
section z'  and  j'  are  the  horizontal  projections  of  the  points  sought. 

Required  to  represent  the  section  which  would  be  made  in  this  ring  by  a 
plane,  N'  T',  parallel  to  the  vertical  plane. 

Such  a  section  will  be  represented  in  its  actual  form  and  dimensions  in  the 
elevation.  To  determine  its  outlines,  let  two  horizontal  planes,  g  q  and  i  k, 
equidistant  from  the  center  line  A  B,  be  supposed  to  cut  the  ring  ;  their  lines 
of  intersection  with  it  will  have  their  horizontal  projections  in  the  two  circles 
g'  o'  and  h'  q1,  which  cut  the  given  plane  N'  T'  in  o'  and  q'.  These  points  being 
projected  vertically  to  0,  q,  k.  etc.,  give  four  points  in  the  curve  required.  The 
line  N'  T'  cutting  the  circle  A'  E'  B'  at  N',  the  projection  N  of  this  point  is 
the  extreme  limit  of  the  curve. 

The  circle  P'  s'j',  the  center  line  of  the  rim  of  the  torus,  is  cut  by  the  planes 
N'  T'  at  the  point  s'9  which,  being  projected  vertically  upon  the  lines  D  P  and 
C  I,  determines  s  and  I,  the  points  of  contact  of  the  curve  with  the  horizontal 


98 


ORTHOGRAPHIC  PROJECTION. 


outlines  of  the  ring.  Finally,  the  points  t  and  u  are  obtained  by  drawing  from 
the  center  0  a  circle,  T'  v',  tangent  to  the  given  plane,  and  projecting  the  point 
of  intersection  v1  to  the  points  v  and  x,  which  are  then  to  be  replaced  upon  C  D 
by  drawing  the  horizontals  v  t  and  x  u. 

Required  to  delineate  the  lines  of  penetration  of  a  sphere  and  a  regular  hex- 
agonal prism  whose  axis  passes  through  the  center  of  the  sphere  (Fig.  189). 


FIG.  189. 

The  centers  of  the  circles  forming  the  two  projections  of  the  sphere  are,  ac- 
cording to  the  terms  of  the  problem,  upon  the  axis  C  0'  of  the  upright  prism, 
which  is  projected  horizontally  in  the  regular  hexagon  D'  E'  F'  G'  H'  I'.  Hence 
it  follows  that,  as  all  the  lateral  faces  of  the  prism  are  equidistant  from  the  cen- 
ter of  the  sphere,  their  lines  of  intersection  with  it  will  necessarily  be  circles  of 
equal  diameters.  The  perpendicular  face,  represented  by  the  line  E'  F'  in  the 
plan,  will  meet  the  surface  of  the  sphere  in  two  circular  arcs,  E  F  and  L  M, 
described  from  the  center  C,  with  a  radius  equal  to  c'  V  or  a'  c'.  And  the  in- 
tersections of  the  two  oblique  faces  D'  E'  and  F'  G'  will  obviously  be  each 
projected  in  two  arcs  of  an  ellipse,  whose  major  axis  d  g  is  equal  to  e1  f,  and 
the  minor  axis  is  the  vertical  projection  of  e'  /'.  But,  as  it  is  necessary  to 


ORTHOGRAPHIC   PROJECTION. 


99 


draw  small  portions  only  of  these  curves,  the  following  method  may  be  em- 
ployed : 

Draw  D  G  through  the  points  E,  F  ;  divide  the  portions  E  F  and  F  G  re- 
spectively into  the  same  number  of  equal  parts,  and,  drawing  perpendiculars 
through  the  points  of  division,  set  off  from  F  G  the  distances  from  the  corre- 
sponding points  in  E  F  to  the  circular  arc  E  C  F,  as  points  in  the  elliptical 
arc  required.  The  remaining  elliptical  arcs  can  be  traced  by  the  same  method. 

Required  to  draw  the  lines  of  penetration  of  a  cylinder  and  a  sphere,  the 
center  of  the  sphere  being  without  the  axis  of  the  cylinder  (Fig.  190). 


FIG.  190. 

Let  the  circle  D'  E'  L'  be  the  projection  of  the  base  of  the  given  cylinder, 
and  let  A  B  be  the  diameter  of  the  given  sphere.  If  a  plane,  as  c'  a",  be  drawn 
parallel  to  the  vertical  plane,  it  will  evidently  cut  the  cylinder  in  two  straight 
lines,  G  G',  H  H'.  This  plane  will  also  cut  the  sphere  in  a  circle  described 
from  the  center  C  with  a  radius  of  half  the  line  c'  d' ;  its  intersection  with  the 
lines  G  G'  and  II  H'  will  give  so  many  points  in  the  curves  sought,  viz., 
G,  H,  I,  K. 

The  planes  a'  l>  and  e'f,  which  are  tangents  to  the  cylinder,  furnish  respect- 


100 


ORTHOGRAPHIC   PROJECTION. 


ively  only  two  points  in  the  curves  ;  of  these  points,  E  and  F  alone  are  visible, 
the  other  two,  L  and  M,  being  concealed  by  the  solid  ;  therefore,  the  planes 
drawn  for  the  construction  of  the  curves  must  be  all  taken  between  a'  V  and 
e'  f.  The  plane  which  passes  through  the  axis  of  the  cylinder  cuts  the  sphere 
in  a  circle  whose  projection  upon  the  vertical  plane  will  meet  at  the  points  D, 
N,  and  g,  Ji,  the  outlines  of  the  cylinder,  to  which  the  curves  of  penetration 
are  tangents. 

To  find  the  lines  of  penetration  of  a  frustum  of  a  cone  and  a  prism 
(Fig.  191). 

The  frustum  is  represented  in  the  plan  by  two  circles  described  from  the 
center  C' ;  and  the  horizontal  lines  M  N"  and  M'  N'  are  the  projections  of  the 


FIG.  191. 

axis  of  a  prism  of  which  the  base  is  square,  and  the  faces  respectively  parallel 
and  perpendicular  to  the  planes  of  projection. 

In  laying  down  the  plan  of  this  solid,  it  is  supposed  to  be  inverted,  in  order 
that  the  smaller  end  of  the  cone  and  the  lines  of  intersection  of  the  lower  sur- 
face, F  G,  of  the  prism  may  be  exhibited.  According  to  this  arrangement,  the 
letters  A'  and  B'  ought,  strictly  speaking,  to  be  marked  at  the  points  I'  and  H', 
and  conversely ;  but,  as  it  is  quite  obvious  that  the  part  above  M'  N'  is  exactly 


ORTHOGRAPHIC  PROJECTION. 


101 


symmetrical  with  that  below  it,  the  distribution  of  the  letters  of  reference 
adopted  in  our  figures  can  lead  to  no  confusion. 

The  intersection  of  the  plane  F  G-  with  the  cone  is  projected  horizontally 
in  a  circle  described  from  the  center  0',  with  the  diameter  F'  G'.  The  arcs 
.!'  F'  A'  and  H'  G'  B'  are  the  only  parts  of  this  circle  which  require  to  be 
drawn.  In  the  vertical  projection  the  extreme  points  K,  L,  A,  B  need  only 
be  found,  for  the  lines  of  intersection  are  here  projected  straight. 

To  describe  the  curves  formed  by  the  intersection  of  a  cylinder  with  the 
frustum  of  a  cone,  the  axes  of  the  two  solids  cutting  each  other  at  right  angles 
(Fig.  192). 


M 


FIG.  192. 


The  projections  of  the  solids  are  laid  down  in  the  figure  precisely  as  in  the 
preceding  example.  The  intersections  of  the  outlines  in  elevation  furnish,  ob- 
viously, four  points  in  the  curves  of  penetration  ;  these  points  are  all  projected 
horizontally  upon  the  line  A'  B'.  Now  pass  a  plane,  as  a  b,  horizontally  through 
both  solids  ;  its  intersection  with  the  cone  will  be  a  circle  of  the  diameter  c  d, 
while  the  cylinder  will  be  cut  in  two  parallel  straight  lines,  represented  in  the 
elevation  by  a  ft,  and  whose  horizontal  projection  may  be  determined  in  the  fol- 
lowing manner:  Conceive  a  vertical  plane,  f  g,  cutting  the  cylinder  at  right 


102 


ORTHOGRAPHIC   PROJECTION. 


angles  to  its  axis,  and  let  the  circle  g  e  f  thereby  formed  be  described  from  the 
intersection  of  the  axes  of  the  two  solids  ;  the  line  j  li  will  now  represent,  in 
this  position  of  the  section,  the  distance  of  one  of  the  lines  sought  from  the 
axis  of  the  cylinder.  Set  oif  this  distance  on  both  sides  of  the  point  A', 
and  through  the  points  k  and  a'  thus  obtained,  draw  straight  lines  parallel  to 
A'  B'  ;  the  intersections  of  these  lines  with  the  circle  drawn  from  the  center  C' 
of  the  diameter  c  d  will  give  four  points  m',  p',  n,  and  o,  which,  being  projected 
vertically  upon  a  b,  determine  two  points,  m  and  p,  in  the  curves  required. 

In  order  to  obtain  the  vertices  or  adjacent  limiting  points  of  the  curves, 
draw  from  the  vertex  of  the  cone  a  straight  line,  t  e,  touching  the  circle  g  e  f, 
and  let  a  horizontal  plane  be  supposed  to  pass  through  the  point  of  contact  e. 
Proceed  according  to  the  method  given  above  to  determine  the  intersections 
of  this  plane  with  each  of  the  solids  in  question,  the  four  points  i'9  r',  q,  and  s, 
which,  being  projected  vertically  upon  the  line  e  r,  determine  the  vertices  i  and 
r  required. 

THE   HELIX. 

The  Helix  is  the  curve 
described  upon  the  surface 
of  a  cylinder  by  a  point  re- 
volving round  it,  and  at  the 
same  time  moving  parallel 
to  its  axis  by  a  certain  in- 
variable distance  during 
each  revolution.  This  dis- 
tance is  called  the  pitch  of 
the  helix  or  screw. 

Required  to  construct 
the  helical  curve  described 
-  by  the  point  A1  upon  a  cyl- 
inder projected  horizontally 
in  the  circle  A'  C'  F',  the 
pitch  being  represented  by 
the  line  A1  A8  (Fig.  193). 

Divide  the  pitcli  A1  A3 
into  any  number  of  equal 
parts,  say  eight;  and 
through  each  point  of  di- 
vision, 1,  2,  3,  etc.,  draw 
straight  lines  parallel  to  the 
ground-line.  Then  divide 
the  circumference  A'  C'  F' 
into  the  same  number  of 
equal  parts  ;  the  points  of 
division,  B',  C',  E',  F',  etc., 
will  be  the  horizontal  pro- 
jections of  the  different  po- 
sitions of  the  given  point 


ORTHOGRAPHIC   PROJECTION. 


103 


during  its  motion  round  the  cylinder.  Thus,  when  the  point  is  at  B'  in  the 
plan,  its  vertical  projection  will  be  the  point  of  intersection  B  of  the  perpen- 
dicular drawn  through  B'  and  the  horizontal  drawn  through  the  first  point  of 
division.  Also,  when  the  point  arrives  at  C'  in  the  plan,  its  vertical  projection 
is  the  point  C,  where  the  perpendicular  drawn  from  C'  cuts  the  horizontal 
passing  through  the  second  point  of  division,  and  so  on  for  all  the  remaining 
points.  The  curve  A1  B  C  F  A3,  drawn  through  all  the  points  thus  obtained, 
is  the  helix  required. 

To  draw  the  vertical  elevation  of  the  solid  contained  between  two  helical  sur- 
faces and  two  concentric  cylinders  (Fig,  193). 

A  helical  surface  is  generated  by  the  revolution  of  a  straight  line  round  the 
axis  of  a  cylinder,  its  outer  end 
moving  in  a  helix,  and  the  line 
itself  forming  with  the  axis  a 
constant  and  invariable  angle. 

Let  A'  C'  F'  and  K'  M'  0' 
represent  the  concentric  bases 
of  the  cylinders,  whose  com- 
mon axis  S  T  is  vertical ;  the 
curve  of  the  exterior  helix  A1 
C  F  A3  is  the  first  to  be  drawn 
according  to  the  method  above 
shown.  Then,  having  set  off 
from  A1  to  A2  the  thickness 
of  the  required  solid,  draw 
through  A2  another  helix 
equal  and  similar  to  the  for- 
mer. Now  construct,  as  above, 
similar  helices,  K  C  0  and  K2 
C2  O2,  of  the  same  pitch  as  the 
last,  but  on  the  interior  cylin- 
der. The  lines  A'  K',  B'  L', 
C'  M',  etc.,  represent  the  hori- 
zontal projections  of  the  va- 
rious positions  of  the  generat- 
ing straight  line,  which,  in 
the  present  example,  has  been 
supposed  to  be  horizontal  ; 
and  these  lines  are  projected 
vertically  at  A1  K,  B  L,  etc. 

It  will  be  observed  that  in 
the  position  A1  K  the  generat- 
ing line  is  projected  in  its  Flo>  194 
actual  length,  and  that  at  the 

position  C'  M'  its  vertical  projection  is  the  point  C.  The  same  remark  applies 
to  the  generatrix  of  the  second  helix.  The  parts  of  both  curves  which  are 
visible  in  the  elevation  may  be  easily  determined  by  inspection. 


104  ORTHOGRAPHIC  PROJECTION. 

To  determine  the  vertical  projection  of  the  solid  formed  by  a  sphere  moving 
in  a  helical  curve  (Fig.  194). 

Let  A'  C'  E'  be  the  base  of  a  cylinder,  upon  which  the  center  point  C'  of  a 
sphere  whose  radius  is  a'  C'  describes  a  helix,  which  is  projected  on  the  vertical 
plane  in  the  curve  A  C  E  J,  determined  as  before.  From  the  various  points 

A,  B,  C,  D ,  in  this  curve,  as  centers,  describe  circles  with  the  radius 

a'  C'  ;  these  denote  the  various  positions  of  the  sphere  during  its  helical  mo- 
tion ;  and,  if  lines  be  drawn  touching  them,  the  curves  thereby  formed  will 
constitute  the  figure  required.  One  of  these  curves  disappears  at  0,  but  reap- 
pears again  at  I.  The  exterior  and  interior  circles  of  the  plan  represent  the 
horizontal  projection  of  the  solid  in  question. 

The  conical  helix  differs  from  the  cylindrical  one  in  that  it  is  described  on 
the  surface  of  a  cone  instead  of  on  that  of  a  cylinder  ;  but  the  construction 
differs  but  slightly  from  the  one  described.  By  following  out  the  same  prin- 
ciples, helices  may  be  represented  as  lying  upon  spheres  or  upon  any  other 
surfaces  of  revolution. 

In  the  arts  are  to  be  found  numerous  practical  applications  of  the  helical 
curve,  as  wood  and  machine  screws,  gears,  and  staircases,  the  construction  of 
which  will  be  still  further  explained  under  their  appropriate  heads. 

DEVELOPMENT   OF    SURFACES. 

The  development  of  the  surface  of  a  solid  is  the  drawing  or  unrolling  on  a 
plane  the  form  of  its  covering  ;  and  if  that  form  be  cut  out  of  paper,  it  would 
exactly  fit  and  cover  the  surface  of  the  solid.  Frequently  in  practice,  the  form 
of  the  surface  of  a  solid  is  found  by  applying  paper  or  thin  sheet-brass  directly 
to  the  solid,  and  cutting  it  to  fit.  Tin  and  copper  smiths,  boiler-makers,  etc., 
are  continually  required  to  form  from  sheet-metal  forms  analogous  to  solids ; 
to  execute  which  they  should  be  able  to  construct  geometrically  the  develop- 
ment of  the  surface  of  which  they  are  to  make  the  form. 

The  development  of  the  surface  of  a  cylinder  is  evidently  but  a  flat  sheet, 
of  which  the  circumference  is  one  dimension  while  its  length  is  the  other. 

To  develop  the  surface  of  a  cylinder  formed  by  the  intersection  of  another 
equal  cylinder,  as  the  knee  of  a  stove-pipe  (Fig.  195). 

Let  A  B  C  D  be  the  elevation  of  the  pipe  or  cylinder.  Above  A  B  describe 
the  semicircle  A'  4'  B'  of  the  same  diameter  as  the  pipe  ;  divide  this  semicir- 
cle into  any  number  of  equal  parts,  eight  for  instance  ;  through  these  points, 
1',  2',  3',  etc.,  draw  lines  parallel  to  the  side  A  0  of  the  pipe,  and  cutting  the 
line  C  D  of  the  intersection  of  the  two  cylinders.  Lay  off  A"  B"  equal  to  the 
semicircle  A'  4'  B',  and  divided  into  the  same  number  of  equal  parts  ;  through 
these  points  of  division  erect  perpendiculars  to  A"  B",  and  on  these  perpen- 
diculars lay  off  the  distances  A"  C",  1"  1",  2"  2",  3"  3",  and  so  on,  corresponding 
to  A  C,  1 1,  2  2,  3  3,  etc.,  in  preceding  figure.  Through  the  points  C",  1",  2", 

D",  draw  connecting  lines,  and  we  have  the  developed  surface  required. 

It  is  to  be  remarked  that  this  gives  but  one  half  of  the  surface  of  the  pipe,  the 
other  being  exactly  similar  to  it. 

To  develop  the  surface  of  a  cylinder  intersected1  by  another  cylinder,  as  in 
the  formation  of  a  "[-pipe  (Fig.  196). 


ORTHOGRAPHIC  PROJECTION. 


105 


7     (!       5         4         3       2     1  A 


7"         6"        5"        4" 3"         2"        1"      A" 


B*     T       6"       5"       A 

r     3"     £"     /"    ^4 

y^ 

Ss 

6 

X 

5 

\ 

FIG.  195. 


The  construction  is  similar  to  the 
preceding,  and,  as  the  same  letters 
and  figures  are  preserved  relatively, 
the  demonstration  will  be  easily  un- 
derstood from  the  foregoing. 

To  develop  the  surface  of  a  right 
cone  (Fig.  197). 

From  C'  as  a  center,  with  a  ra- 
dius, C'  A',  equal  to  the  inclined 
side  A  C  of  the  cone,  describe  an  arc 
of  a  circle  A'  B'  A" ;  on  this  arc  lay 
off  the  distance  A'  B'  A"  equal  to 
the  circumference  of  the  base  of  the 
cone  ;  connect  A'  0'  and  C'  A",  and 
A'  B'  A"  C'  is  the  developed  surface  required. 

To  develop  the  surface  of  the  frustum  of  a  cone,  D  A  B  E  (Fig.  197). 
D'  E'  D"  is  the  development  of  the  cut-off  cone  C  D  E  as  shown  by  the  pre- 
ceding construction,  and  we  therefore  have  A'  B'  A"  D"  E'  D  as  the  developed 
surface  of  the  frustum. 

To  develop  the  surface  of  a  frustum  of  a  cone,  when  the  cutting  plane  a  h 
is  inclined  to  the  base  (Fig.  197). 

On  A  B,  the  base,  describe  the  semicircle  A  3'  B  ;  divide  the  semicircle  into 
any  number  of  equal  parts,  six  for  instance  ;  from  each  point  of  division,  1',  2', 
3',  4',  5',  let  fall  perpendiculars  to  the  base  ;  at  1,  2,  3,  4,  5,  connect  each  of 
these  last  points  with  the  apex  0.  Divide  now  the  arc  A'  B'  A*,  equal  to  the 
base  A  3  B,  into  twelve  equal  parts  ;  each  of  these  parts  by  the  construction 
is  equal  to  the  arc  A  1',  V  2' ;  connect  these  points  of  division  with  the 
point  C' ;  on  C'  A'  take  C'  a'  equal  to  C  a,  a  being  the  point  at  which  the 


106 


ORTHOGRAPHIC   PROJECTION. 


E 


plane  cuts  the  inclined  side  of  the  cone  ;  in  the  same  way  on  C'  B',  lay  off 

C'  b'  equal  to  C  b. 

It  is  evident  that  all  the  lines  connecting  the  apex  C  with  the  base,  included 

within  the  two  inclined  sides,  are  rep- 
resented as  less  than  their  actual  length, 
and  must  be  projected  on  the  inclined 
sides  to  determine  their  absolute  di- 
mensions ;  project,  therefore,  the  points 
1",  2",  3",  4",  5",  at  which  the  cutting 
plane  intersects  the  lines  C  1,  C  2,  C  3, 
04,  05,  by  drawing  parallels  to  the 
base  through  these  points  to  the  in- 
clined side  C  B.  Now  lay  off  C'  1"", 
0'  2"",  etc.,  equal  to  C  1'",  C  2",  etc.; 
connect  the  points  a',  I"",  2"",  -  -  b1, 
]  A a",  and  we  have  the  developed  sur- 
face a'  A'  B'  A"  a"  b'  required. 


A 


B' 


FIG.  198. 


FIG.  199. 


ORTHOGRAPHIC   PROJECTION. 


107 


To  develop  the  surface  of  a  sphere  or  ball  (Figs.  198,  199). 

It  is  evident  that  the  surface  can  not  be  accurately  represented  on  a  plane. 
It  may  be  done  approximately  by  a  number  of  gores.  Let  CAB  (Fig.  199) 
be  the  eighth  of  a  hemisphere  ;  on  C  D  describe  the  quarter  circle  D  A  c ; 
divide  this  arc  into  any  number  of  equal  parts,  six  for  instance  ;  from  the  points 
of  division  1,  2,  3,  ...  let  fall  perpendiculars  on  C  D,  and  from  the  intersec- 
tions with  this  line  describe  arcs  1'  1",  2'  2",  3'  3",  .  .  .  cutting  the  line  C  B  at 

1",  2",  3", ;  on  the  straight  line  C'  D'  (Fig.   198),  lay  off  C'  D'  equal  to 

the  arc  D  A  c,  with  as  many  equal  divisions  ;  then 
from  either  side  of  this  line  lay  off  1'"  1"",  2'"  2"" 

D'  B'  equal  to  the  arcs  1'  1',  2'  2" D  B 

(Fig.  199).     Connect  the  points  C',  1"",  2™, and 

C'  A'  B'  is  approximately  the  developed  surface. 

It  is  to  be  remarked  that,  in  the  preceding  demon- 
strations, the  forms  are  described  to  cover  the  surface 
only ;  in  construction,  allowance  is  to  be  made  for 
lap  by  the  addition  of  margins  on  each  side  as  neces- 
sary. It  is  found  difficult,  in  the  formation  of  hemi- 
spherical ends  of  boilers,  to  bring  all  the  gores  to- 
gether at  the  apex  ;  it  is  usual,  therefore,  to  make  them,  as  shown  (Fig.  200), 
by  cutting  short  the  gores,  and  surmounting  the  center  with  a  cap-piece. 

SHADE-LINES. 

In  outline  drawings,  or  drawings  which  consist  simply  of  the  lines  employed 
to  indicate  the  form  of  the  object  represented,  the  roundness,  the  flatness,  or 
the  obliquity  of  individual  surfaces,  is  not  indicated  by  the  lines,  although  it 
may  generally  be  inferred  from  the  relation  of  different  views  of  the  same  part. 
The  direct  significance  of  an  out- 

\ 


FIG.  200. 


line  drawing  may,  however,  be  con- 
siderably increased,  by  strengthen- 
ing those  lines  which  indicate  the 
contours  of  surfaces  resting  in  the 
shadow  ;  and  this  distinction  also 
improves  the  general  appearance  of 
the  drawing.  The  strong  lines,  to 
produce  the  best  effect,  ought  to 
be  laid  upon  the  sharp  edges  at 
the  summits  of  salient  angles  ;  but 
bounding  lines  for  curve  surfaces 
should  be  drawn  finely,  and  should 
be  but  slightly,  if  at  all,  strength- 
ened on  the  shade  side.  This  dis- 
tinction assists  in  contrasting  flat 
and  curve  surfaces.  To  understand 
and  apply  the  shade-lines,  however,  we  must  know  the  direction  in  which 
the  light  is  supposed  to  fall  upon  the  object,  and  thence  the  locality  of  the 
shadows. 


FIG.  201. 


108 


ORTHOGRAPHIC  PROJECTION. 


It  is  necessary,  for  the  explicitness  of  the  drawing,  that,  firstly,  the  light  be 
supposed  to  fall  upon  the  object  in  parallel  lines,  that  all  the  parts  may  be 
shade-lined  according  to  one  uniform  rule  ;  secondly,  that  the  light  should  be 
supposed  to  fall  upon  the  object  obliquely,  as  in  this  way  both  the  horizontal 
and  vertical  lines  may  be  relieved  by  shading. 

Fig.  201  represents  the  drawing  of  a  cube,  with  its  projections  on  a  vertical 


FIG.   202. 


FIG.  203. 


and  on  a  horizontal  plane,  or  in  elevation  and  plan,  all  in  perspective.  The 
arrows  show  the  directions  in  which  the  light  is  supposed  to  fall  :  in  space 
diagonally  through  the  body  of  the  cube  and  in  projection  diagonally  through 
the  squares  representing  the  plan  and  elevation  of  the  cube.  The  projections 
of  the  rays,  therefore,  form  -angles  of  45°  with  the  ground-line,  which  line  is 
represented  in  the  figure  by  B  D.  In  the  old  method,  still  used  in  topograph- 
ical and  by  many  in  mechanical  drawings,  the  light  is  supposed  to  fall  in  space, 
as  if  A  D  were  the  ground-line,  but  the  shade-lines  in  the  vertical  plane  are  the 
same  in  both  methods. 

Copies  of  a  few  of  the  preceding  projections  are  here  given,  with  the 
proper  shade-lines,  according  to  the  first  or  French  method  (Fig.  201).  The 
outlines  to  be  shaded  can  be  determined,  ordinarily,  by  mere  inspection  and  by 
using  a  45°  triangle.  Such  a  triangle  gives  the  direction  of  the  projected  rays, 


ORTHOGRAPHIC  PROJECTION. 


109 


and  determines  the  surfaces  in  shadow.     Fig.  202  is  a  reproduction  of  Fig.  171 ; 

Fig.  203  is  a  reproduction  of  Fig.  176  ;  Fig.  204  is  a  reproduction  of  Fig.  184  ; 

Fig.  205  is  a  reproduction  of  the  plan  of  Fig.  188.  The  outlines  on  which 

the  light  falls  are  represented  by  fine  lines,  the 
others  by  coarse  lines.  In  general,  it  is  not  cus- 
tomary to  use  more  than  two  grades  of  lines, 
one  for  the  outlines  in  light,  and  the  other  for 
those  in  shade ;  but,  for  lines  parallel  with  the 


FIG.  204. 


FIG.  205. 


rays  of  light,  medium  lines  are  sometimes  used,  and  sometimes  the  shade-lines 
are  proportioned  to  the  depth  of  the  surfaces  to  which  they  belong,  below  the 
original  surfaces  from  which  the  shadows  arise. 


SHADES    AND    SHADOWS. 

LIGHT  is  diffused  through  space  in  straight  lines,  and  the  lines  of  light  are 
called  ray*.  When  the  source  of  light  is  situated  at  a  very  great  distance  from 
the  illuminated  objects,  as  in  the  case  of  the  sun  with  relation  to  the  earth,  the 
rays  of  light  do  not  sensibly  diverge,  and  may  be  regarded  as  exactly  parallel  to 
each  other.  Such  is  the  case  in  mechanical  drawings,  where  the  objects  to  be 
represented  are  always  regarded  as  illuminated  by  the  solar  light. 

Light  is  called  direct  when  it  is  transmitted  to  an  object  without  the  inter- 
vention of  any  opposing  medium.  But,  as  all  bodies  subjected  to  the  action  of 
light  possess,  in  a  greater  or  less  degree,  the  property  of  giving  out  a  certain 
portion  of  it  to  the  surrounding  objects, this  reflected  light  becomes  in  its  turn, 
though  with  greatly  diminished  intensity,  a  source  of  illumination  to  those 
objects  which  are  deprived  of  direct  light. 

Everything  which  tends  to  intercept  or  prevent  the  direct  light  from  falling 
upon  a  body,  produces  upon  the  surface  of  that  body  a  degree  of  obscurity 
of  greater  or  less  intensity  ;  this  is  called  a  shade  or  shadow.  Such  effects  are 
usually  classified  as  direct  shadows  and  cast  shadows. 

The  shade  proper,  or  direct  shadow,  is  that  which  occurs  on  that  portion  of 
the  surface  of  a  body  which  is  situated  opposite  to  the  enlightened  part,  and  is 
the  natural  result  of  the  form  of  the  body  itself,  and  of  its  position  with  regard 
to  the  rays  of  light.  The  cast  shadow,  on  the  other  hand,  is  that  which  is 
produced  upon  the  surface  of  one  body  by  the  interposition  of  another  between 
the  former  and  the  source  of  light  ;  thus  intercepting  the  rays  which  would 
otherwise  illuminate  that  surface.  Cast  shadows  may  also  obviously  be  pro- 
duced upon  the  surface  of  a  body  by  the  form  of  the  body  itself  ;  as,  for  exam- 
ple, if  it  contain  projecting  or  concave  parts. 

The  limit  of  the  direct  shadow  on  any  body,  whatever  may  be  its  form  or 
position,  is  a  line  of  greater  or  less  distinctness,  termed  the  line  of  shade  ;  this 
line  is,  of  course,  determined  by  the  contact  of  the  luminous  rays  with  the 
surface  of  the  body  ;  .and,  if  these  rays  be  prolonged  till  they  meet  a  given  sur- 
face, by  joining  all  the  points  of  intersection  with  that  surface,  we  obtain  the 
outline  of  the  shadoiv  cast  upon  it  by  the  part  of  the  body  which  is  deprived  of 
light. 

The  rays  of  light  being  regarded  as  parallel  to  each  other,  it  is  obvious  that, 
in  the  delineation  of  shadows,  it  is  only  necessary  to  know  the  direction  of  one 
of  them  ;  and,  as  that  direction  is  arbitrary,  we  have  adopted  the  usual  and 
confessedly  the  most  convenient  mode  of  regarding  the  rays  as  in  all  cases 
falling  in  the  direction  of  the  diagonal  of  a  cube,  of  which  the  sides  are  parallel 


SHADES   AND   SHADOWS. 


Ill 


to  the  planes  of  projection.  This  is  graphically  shown  in  Fig.  201  of  the  pre- 
ceding chapter.  The  projections  of  the  ray  form  each  an  angle  of  45°  with 
the  ground-line.  This  is  not  true  of  the  ray  itself  in  space,  for  that  forms 
an  angle  of  54°  44'  with  the  ground-line,  and  an  angle  of  35°  16'  with  each  of 
the  planes  of  projection. 

To  find  the  shadow  of  a  point,  as  A,  A'  (Fig.  206),  on  either  plane  of  pro- 
jection, the  vertical,  for  instance,  we  draw  a  line  through  the  horizontal  pro- 
jection of  the  point  A'  at  an  angle  of  45°  with  the  ground-line,  and  at  the 
point  of  intersection  of  those  lines,  a',  erect  a  perpendicular  to  intersect  the 
vertical  projection  of  the  ray  through  A,  which  will  be  at  the  point  a,  the 
shadow  in  question. 

This,  as  may  readily  be  seen,  is  simply  finding  the  point  of  intersection  of 
the  ray  passing  through  the  point  and  the  vertical  plane  of  projection.  The 
converse  of  this  method  will  as  easily  determine  the  shadow  of  the  point  on 
the  horizontal  plane. 

The  line  A  a  in  the  elevation  being  equal  in  every  case  to  the  line  A'  a'  in 
the  plan,  it  will  in  some  cases  be  found  more  convenient  to  use  the  compasses 
instead  of  a  geometrical  construction  ;  for  example,  in  place  of  projecting  the 
point  a'  by  a  perpendicular  to  the  ground-line,  in  order  to  obtain  the  position 
of  the  required  shadow  a,  that  point  may  be  found  by  simply  setting  off  upon 
the  line  A  a  a  distance  equal  to  A'  a'. 

In  the  following  illustrations  the  same  letter  accented  is  employed  in  the 
plan  as  in  the  elevation  to  refer  to  the  same  object  or  point. 

Required  to  determine  the  shadow  cast  upon  the  vertical  wall  X  Y  by  the 
straight  line  A  B  (Fig.  206). 

It  is  obvious  that  in  this  case  the  shadow  itself  will  be  a  straight  line  ; 
hence,  to  solve  the  problem,  it  is  only  necessary  to  find  two  points  in  that  line. 
We  have  seen  that  the  position  of  the  shadow  thrown  by  the  point  A  is  at  a; 


lillllillililiilliiiililiillllilllllllll       liliillllillllllllllllll 


FIG.  206. 


by  a  similar  process  we  can  easily  determine  the  point  #,  the  position  of  the 
shadow  thrown  by  the  opposite  extremity  B  of  the  given  line  ;  the  straight 
line  a  b,  which  joins  these  two  points,  is  the  shadow  required. 

It  is  evident,  from  the  construction  of  this  figure,  that  the  line  a  b  is  equal 


112 


SHADES  AND   SHADOWS. 


and  parallel  to  the  given  line  A  B  ;  this  results  from  the  circumstance  that  the 
latter  is  parallel  to  the  vertical  plane  X  Y.  Hence,  when  a  line  is  parallel  to  a 
plane,  its  shadow  upon  that  plane  is  a  line  ivhich  is  equal  and  parallel  to  it. 

Suppose  now  that,  instead  of  a  mere  line,  a  parallel  slip  of  wood  or  paper, 
A  B  C  D,  be  taken,  which,  for  the  sake  of  greater  simplicity,  we  shall  conceive 
as  having  no  thickness.  The  shadow  cast  by  this  object  upon  the  same  verti- 
cal plane  X  Y  is  a  rectangle  a  b  c  d,  equal  to  that  which  represents  the  projec- 
tion of  the  slip,  because  all  the  edges  of  the  latter  are  parallel  to  the  plane  upon 
which  the  shadow  is  thrown.  Hence,  in  general,  when  any  surface,  whatever 
may  be  its  form,  is  parallel  to  a  plane,  its  shadow  thrown  upon  that  plane  is  a 
figure  similar  to  it,  and  similarly  situated.  This  principle  facilitates  the  de- 
lineation of  shadows  in  many  cases.  In  the  present  example,  an  idea  may 
be  formed  of  its  utility  ;  for,  after  having  determined  the  position  of  any  one 
of  the  points  a,  b,  c,  d,  the  figure  may  be  completed  by  drawing  lines  equal  and 
parallel  to  the  sides  of  the  slip,  without  requiring  to  go  through  the  operations 
in  detail. 

When  the  object  is  not  parallel  to  the  given  plane,  the  shadow  cast  is  no 
longer  a  figure  equal  and  similarly  placed  ;  the  method  of  determining  it  re- 
mains, however,  unchanged  ;  thus  (Fig.  207),  take  the  portion  A  E  of  the  slip 
A  B,  which  throws  its  shadow  on  the  plane  X  Y  ;  draw  the  projections  of  the 
rays  of  light  A  a,  E  e,  C  c,  F/,  and  A'  a'.  E'  e',  and  project  a'  vertically  to  a,  c, 
and  e'  to  e,  f ;  connect  a,  e,  f,  c,  and  we  have  the  outline  of  the  shadow  of  the 
slip  A  E. 

By  an  exactly  similar  construction  we  have  the  shadow  of  the  portion  E  B 
on  the  plane  Y  Z,  which,  being  inclined  to  the  plane  of  projection  in  a  direction 
contrary  to  X  Y,  necessarily  causes  the  shadow  to  be  broken,  and  the  part  e  d 
to  lie  in  a  contrary  direction  to  af. 

The  determination  of  the  shadow  of  the  slip  upon  a  molding  placed  on 
the  plane  X  Y  parallel  to  the  slip  (Fig.  208)  can  be  readily  determined  by  an 
inspection  of  the  figure. 


E' 


FIG.  210. 


FIG.  211. 


When  the  slip  is  placed  perpendicularly  to  a  given  plane,  X  Y  (Fig.  209),  on 
which  a  projecting  molding,  of  any  form  whatever,  is  situated,  the  shadow  of 
the  upper  side  A'  B',  which  is  projected  vertically  in  A,  will  be  simply  a  line, 


SHADES   AND   SHADOWS. 


113 


A  a,  at  an  angle  of  45°,  traversing  the  entire  surface  of  the  molding,  and  pro- 
longed unbroken  beyond  it.  This  may  easily  be  demonstrated  by  finding  the 
position  of  the  shadow  of  any  number  of  points  such  as  D',  taken  at  pleasure 
upon  the  straight  line  A'  B'.  The  shadow  of  the  opposite  side,  projected  in  C, 
will  follow  the  same  rule,  and  be  denoted  by  the  line  C  c,  parallel  to  the 
former.  Hence,  as  a  useful  general  rule  :  in  all  cases  where  a  straight  line  is 
perpendicular  to  a  plane  of  projection,  it  throws  a  shadow  upon  that  plane 
in  a  straight  line,  forming  an  angle  of  45°  ivith  the  ground-line. 

When  the  slip  is  set  horizontally  in  reference  to  its  own  surface,  and  per- 
pendicularly to  the  given  plane  X  Y  (Fig.  210),  the  shadow  commences  from 
the  side  D  B,  which  is  in  contact  with  this  plane,  and  terminates  in  the  hori- 
zontal line  a  c,  which  corresponds  to  the  opposite  side  A  C  of  the  slip. 

To  find  the  shadow  cast  by  a  slip,  A  B  C  D,  upon  a  curved  surface,  either 
convex  or  concave,  whose  horizontal  projection  is  represented  by  the  line  X  e'  Y 
(Fig.  211). 

This  construction  is  similar  to  the  foregoing  illustrations,  and  requires  no 
explanation  more  than  the  figure. 

Required  to  find  the  shadow  cast  upon  a  vertical  plane,  X  Y,  by  a  given 
circle  parallel  to  it  (Fig.  212). 

Let  C,  C',  be  the  projections  of  the  center  of  the  circle,  and  R,  R/,  those  of 
the  rays  of  light. 

The  position  of  the  shadow  of  the  center  C,  according  to  the  rules  already 
fully  developed,  is  easily  fixed  at  c  ;  from  which  point,  if  a  circle  equal  to  the 
given  circle  be  described,  it  will  represent  the  outline  of  the  required  shadow, 
according  to  the  principle  previously  enunciated  on  page  112. 

When  the  circle  is  perpendicular  to  both  planes  of  projection  (Fig.  213),  its 


Li' 


FIG.  213. 


D' 
FIG.  214. 


projection  upon  each  will  obviously  be  represented  by  the  equal  diameters  A  B 
and  0'  D',  perpendicular  each  to  the  ground-line.  To  determine  the  cast 
shadow,  describe  the  given  circle  upon  both  planes,  as  indicated  in  the  figures, 
and  divide  the  circumference  of  each  into  any  number  of  equal  parts  ;  then, 
having  projected  the  points  of  division,  as  Aa,  E2,  C2,  etc.,  to  their  respective 
8 


114 


SHADES   AND   SHADOWS. 


diameters  A  B  and  C'  D',  draw  from  them  lines  parallel  to  the  rays  of  light, 
which,  by  their  intersection  with  the  given  plane,  will  indicate  so  many  points 
in  the  outline  of  the  cast  shadow. 

If  the  given  circle  be  horizontal  (Fig.  214),  its  shadows  cast  upon  the 
straight  and  curved  portions  of  the  vertical  plane  X  Y  become  ellipses,  which 
must  be  constructed  by  means  of  points,  as  indicated  in  the  figure. 

If  the  plane  of  the  circle  is  situated  perpendicularly  to  the  vertical  projec- 
tion of  the  luminous  rays  (Fig.  215),  the  method  of  constructing  the  cast 
shadow  does  not  differ  from  that  pointed  out  in  reference  to  Fig.  214.  It  is 
obvious  that,  instead  of  laying  down  the  entire  horizontal  projection  of  this 
circle,  all  that  is  necessary  is  to  set  off  the  diameter  D'  E'  equal  to  A  B,  be- 
cause the  shadow  of  this  diameter,  transferred  in  the  usual  way,  gives  the 
major  axis  of  the  ellipse  which  constitutes  the  outline  of  the  shadow  sought, 
while  its  minor  axis  is  at  once  determined  by  a  b,  equal  and  parallel  to  A  B. 

To  delineate  the  shadow 

of  a  cirde  paralhl  to  the 

vertical  plane  of  projection, 
throwing  its  shadow  at  once 
upon  two  plane  surfaces  in- 
clined to  each  other  (Fig. 
216),  all  that  it  is  neces- 
sary specially  to  point  out 
is,  that  the  points  d  and  e 
are  found  by  drawing  from 
Y  a  line  Y  D',  parallel  to 
the  rays  of  light,  and  pro- 
jecting the  point  D'  to  D 
andE. 

We    may  here   remark 
that,  in  every  drawing 

where  the  shadows  are  to  be  inserted,  it  is  of  the  utmost  importance  that  the 
projections  which  represent  the  object  whose  shadow  is  required  should  be 
exactly  defined,  as  well  as  the  surface  upon  which  this  shadow  is  cast  ;  it  is 
therefore  advisable,  in  order  to  prevent  mistakes  and  to  insure  accuracy,  to 
draw  the  figures  in  India  ink,  and  to  erase  all  pencil-marks  before  proceeding 
to  the  operations  necessary  for  finding  the  shadows. 

To  find  the  outline  of  the  shadow  cast  upon  both  planes  of  projection  by  a 
regular  hexagonal  pyramid  (Fig.  217). 

It  is  obvious  that  the  three  sides  A'  B'  F,  A'  B'  C',  and  A'  C'  D',  alone 
receive  the  light  ;  consequently  the  edges  A'  F'  and  A'  D'  are  the  lines  of  shade. 
To  solve  this  problem,  then,  we  have  only  to  determine  the  shadow  cast  by 
these  two  lines,  which  is  accomplished  by  drawing  from  the  projections  of  the 
vertex  of  the  pyramid  the  lines  A  b'  and  A'  a'  parallel  to  the  ray  of  light,  then 
raising  from  the  point  b'  a  perpendicular  to  the  ground-line,  which  gives  at  a' 
the  shadow  of  the  vertex  on  the  horizontal  plane  (on  the  other  side  of  the 
ground-line),  and  finally  by  joining  this  last  point  a'  with  the  points  D'  and  F' ; 
the  lines  D'  a'  and  F'  a'  are  the  outlines  of  the  required  shadow  on  the  hori- 


FIG.  215. 


ir 

FIG.  216. 


SHADES   AND   SHADOWS. 


115 


2ontal  plane.  But,  as  the  pyramid  happens  to  be  situated  sufficiently  near  the 
vertical  plane  to  throw  a  portion  of  its  shadow  toward  the  vertex  upon  it,  this 
portion  may  be  found  by  raising  from  the  point  c,  where  the  line  A'  a'  cuts  the 
ground-line,  a  perpendicular  c  a,  intersecting  the  line  A  V  in  a  ;  the  lines  a  d 
and  a  e  joining  this  point  with  those  where  the  horizontal  part  of  the  shadow 
meets  the  ground-line,  will  be  its  outline  upon  the  vertical  plane. 


FIG.  217. 


FIG.  218. 


To  determine  the  limit  of  shade  on  a  cylinder  placed  vertically,  and  likewise 
its  shadow  cast  upon  the  two  planes  of  projection  (Fig.  218). 

The  lines  of  shade  on  a  cylinder  situated  as  indicated  are  at  once  found  by 
drawing  two  tangents  to  its  base,  parallel  to  the  ray  of  light,  and  vertically 
projecting  through  the  points  of  contact  lines  parallel  to  the  axis  of  the  cyl- 
inder. 

Draw  the  tangents  D'  d'  and  C'  c'  parallel  to  the  rays  of  light  ;  these  are  the 
outlines  of  the  shadow  cast  upon  the  horizontal  plane.  Through  the  point  of 
contact  C'  draw  the  vertical  line  C  C'  ;  this  line  denotes  the  line  of  shade  upon 
the  surface  of  the  cylinder.  It  is  obviously  unnecessary  to  draw  the  perpen- 
dicular from  the  opposite  point  D',  because  it  is  altogether  concealed  in  the 
vertical  elevation  of  the  solid.  In  order  to  ascertain  the  points  C'  and  D'  with 
accuracy,  draw  through  the  center  0'  a  diameter  perpendicular  to  the  rays  of 
light. 

Had  this  cylinder  been  placed  at  a  somewhat  greater  distance  from  the 
vertical  plane  of  projection,  its  shadow  would  have  been  entirely  cast  upon  the 
horizontal  plane,  in  which  case  it  would  have  terminated  in  a  semicircle  drawn 
from  the  center  o',  with  a  radius  equal  to  that  of  the  base.  But,  as  a  portion 
of  the  shadow  of  the  upper  part  is  thrown  upon  the  vertical  plane,  its  outline 
will  be  denned  by  an  ellipse  drawn  in  the  manner  indicated  in  Fig.  214. 

To  find  the  line  of  shade  in  a  reversed  cone,  and  its  shadow  cast  upon  the 
two  planes  of  projection  (Fig.  219). 


116 


SHADES  AND   SHADOWS. 


From  the  center  A'  of  the  base  draw  a  line  parallel  to  the  ray  of  light ; 
from  the  point  a'  where  it  intersects  the  perpendicular,  describe  a  circle  equal 
to  the  base,  and  from  the  point  A'  draw  the  lines  A'  b'  and  A'  c'9  tangent  to 
this  circle  ;  these  are  the  outlines  of  the  shadow  cast  upon  the  horizontal  plane. 
Then  from  the  center  A'  draw  the  radii  A'  B'  and  A'  C'  parallel  to  a'  V  and 
a'  c'  ;  these  radii  are  the  horizontal  projections  of  the  lines  of  shade,  the  former 
of  which,  transferred  to  B  D,  is  alone  visible  in  the  elevation.  But,  in  order 
to  complete  the  outline  of  the  shadow,  it  is  necessary  to  project  the  point  C'  to 
C,  from  which,  by  a  construction  which  will  be  manifest  by  inspecting  the 
figures,  we  derive  the  point  c  and  the  line  c  d  as  part  of  the  cast  shadow  of  the 
line  C'  A'.  The  rest  of  the  outline  of  the  vertical  portion  of  the  cast  shadow 
is  derived  from  the  circumference  of  the  base,  as  in  Fig.  218. 

To  find  the  line  of  shade  and  the  shadow  of  a  horizontal  cylinder  inclined  to 
the  vertical  plane  (Fig.  220). 


FIG.  219. 


FIG.  220. 


The  construction  in  this  case  is  the  same  as  that  explained  by  Fig.  218.  Of 
the  horizontal  lines  of  shade  A  B  and  C  D,  the  latter  alone  is  visible  in  the 
elevation,  while,  on  the  other  hand,  the  former,  A  B  alone,  is  seen  in  the  plan, 
where  it  may  be  found  by  drawing  a  perpendicular  from  A  meeting  the  base 
F'  G'  in  A'.  The  line  A'  E',  drawn  parallel  to  the  axis  of  the  cylinder,  is  the 
line  of  shade  required.  Project  the  shadow  of  the  line  A  B  on  the  vertical 
plane  as  in  previous  examples,  and  the  construction  will  define  the  outline  of 
the  shadow  of  the  cylinder. 

The  example  here  given  presents  the  particular  case  in  which  the  bases  of 
the  cylinder  are  parallel  to  the  direction  of  the  rays  of  light.  In  this  case,  to 
determine  the  line  A'  E',  lay  off  the  angle  A'  L  A2  equal  to  35°  16',  which  the 
ray  of  light  makes  with  the  horizontal  plane,  so  that  the  side  Aa  L  shall  be 
tangent  to  the  circle  F'  A2  G'  (which  represents  the  base  of  the  cylinder  laid  down 
on  the  horizontal  plane)  ;  through  the  point  of  tangency  A2,  draw  a  line,  A'  E', 
parallel  to  the  axis  of  the  cylinder,  which  will  be  the  line  of  shade,  as  before. 


SHADES  AND   SHADOWS. 


117 


To  determine  the  shadows  cast  upon  a  cylinder  by  various  shaped  caps. 

Fig.  221  represents  a  cylinder  upon  which  a  shadow  is  thrown  by  a  rec- 
tangular prism,  of  which  the  sides  are  parallel  to  the  planes  of  projection. 
The  shadow  in  this  case  is  derived  from  the  edges  A'  D'  and  A'  E',  the  first 
of  which,  being  perpendicular  to  the  plane  of  projection,  gives,  according  to 
principles  already  laid  down,  a  straight  line  at  an  angle  of  45°  for  the  outline 
of  its  shadow,  whereas  the  side  A'  E'  being  parallel  to  that  plane,  its  shadow 
is  determined  by  a  portion  of  a  circle,  a  b  c,  described  from  the  center  o. 

If  the  prism  be  hexagonal  (Fig.  222),  or  a  cylinder  be  substituted  for  it 


J)'~- 


7?r/.      for 
Jj'  •«  6\ 

.        K 

'     --J     \ 

\----r 

\ 

C'\ 

V  

(A 


FIG.  221. 


FIG.  222. 


FIG.  223. 


(Fig.  223),  the  mode  of  construction  remains  the  same.  But  it  should  be 
observed  that  it  is  best  in  all  such  cases  to  commence  by  finding  the  points 
which  indicate  the  main  direction  of  the  outline.  To  ascertain  the  point  a 
at  which  the  shadow  commences,  draw  from  a'  the  line  a'  A'  at  an  angle  of 
45°,  which  is  then  to  be  projected  vertically  to  a  A.  Then  the  highest  point  Z» 
(Fig.  223)  should  be  determined  by  the  intersection  of  the  radius  0'  B'  (drawn 
parallel  to  the  ray)  with  the  circumference  of  the  base  of  the  cylinder  on 
which  the  required  shadow  is  cast  ;  and,  finally,  the  point  c,  where  the  outline 
of  the  cast  shadow  intersects  the  line  of  shade,  should  be  determined  by  a  simi- 
lar process. 

To  determine  the  shadows  cast  upon  a  hexagonal  prism  ~by  the  same  caps. 

Fig.  224  represents  a  hexagonal  prism  upon  which  a  shadow  is  thrown 
by  a  rectangular  prism. 

Fig.  225  represents  a  hexagonal  prism  upon  which  a  shadow  is  cast  by 
another  hexagonal  prism. 

Fig.  226  represents  a  hexagonal  prism  upon  which  a  shadow  is  cast  by  a 
cylinder.  These  three  cases  do  not  materially 'differ  from  the  preceding  three, 
and  can  easily  be  understood  from  an  examination  of  the  figures. 

To  define  the  shadows  cast  upon  the  interior  of  a  hollow  cylinder,  in  section, 
by  itself,  and  by  a  circular  piston  fitted  into  it  (Fig.  227). 

The  figure  shows  the  section  of  a  steam-cylinder,  by  a  plane  passing  through 
its  axis,  with  its  piston  and  rod  in  full. 

Conceive,  in  the  first  instance,  the  piston  P  to  be  removed ;  the  shadow  cast 


118 


SHADES  AND   SHADOWS. 


into  the  interior  of  the  cylinder  will  then  consist,  obviously,  of  that  projected 
by  the  vertical  edge  B  C,  and  by  a  portion  of  the  horizontal  edge  B  A.  To  find 
the  first,  draw  through  B'  a  line,  B'  V,  at  an  angle  of  45°  with  B'  A' ;  the  point 


D 


\i 


J    e 


FIG.  224. 


FIG.  225. 


FIG.  226. 


hA 


b',  where  this  line  meets  the  interior  surface  of  the  cylinder,  being  projected 
vertically,  gives  the  line  b  f  as  the  outline  of  the  shadow  sought.  Then,  paral- 
lel to  the  direction  of  the  light,  draw  a  tangent  at  F'  to  the  inner  circle  of  the 
base  ;  its  point  of  contact,  being  projected  to  F  in  the  ele- 
vation, marks  the  commencement  of  the  outline  of  the 
shadow  cast  by  the  upper  edge  of  the  cylinder.  The  point 
b,  where  it  terminates,  will  obviously  be  the  intersection 
of  the  straight  line  /  b  already  determined,  with  a  ray,  B  b, 
from  the  upper  extremity  of  the  edge  B  C  ;  and  any  inter- 
mediate point  in  the  curve,  as  e,  may  be  found  in  precisely 
the  same  way.  The  outline  of  the  shadow  required  will 
then  be  the  curve  F  e  b  and  the  straight  line  b  f.  Sup- 
pose, now,  the  piston  P  and  its  rod  T  to  be  inserted  into 
the  cylinder,  as  shown.  The  lower  surface  of  the  piston 
will  then  cast  a  shadow  upon  the  interior  surface  of  the 
cylinder,  of  which  the  outline  D  d  h  o  may  be  formed  in 
the  same  way,  as  will  be  obvious  from  inspection  of  the 
figures  and  comparison  of  the  letters  of  reference.  The 
FIG.  227.  piston-rod  T  being  cylindrical  and  vertical,  it  casts  also 

its  shadow  into  the  interior  of  the  cylinder ;  it  will  obviously  consist  of  the 
rectangle  ij  I  k  drawn  parallel  to  the  axis. 

To  find  the  shadow  cast  in  the  interior  of  a  hollow  cylinder,  surmounted  by 
a  circular  disk  or  cover,  sectioned  through  the  center,  where  it  is  also  penetrated 
by  a  cylindrical  aperture  (Fig.  22S). 

The  construction  necessary  for  finding  the  outlines  of  the  cast  shadow  will 
obviously  be  the  same  as  already  laid  down.  To  know  beforehand  what  parts 
of  the  upper  and  lower  edges  of  the  central  aperture  cast  their  shadows  into 
the  interior  of  the  cylinder,  in  order  to  avoid  unnecessary  work,  we  should  first 
determine  the  position  of  the  point  of  intersection,  c,  of  the  two  curves  b  cf 
and  ace,  shadows  of  these  edges,  which  is  the  cast  shadow  of  the  lowest  point,. 


SHADES  AND   SHADOWS. 


119 


0,  in  the  curve  D  C,  previously  laid  down  in  the  circular  opening  of  the  cover, 
in  the  manner  indicated  in  the  previous  example. 

To  find  the  shadow  cast  in  the  interior  of  a  cylinder,  in  section,  inclined  to 
the  horizontal  plane  (Fig.  229). 

In  any  convenient  part  of  the  paper,  draw  the  diagonal  m  o  parallel  to  the 
line  of  light  A'  E,  and  construct  a  square  m  n  o  p  (Fig.  230)  ;  from  one  of  the 


0' 


extremities,  o,  draw  the  line  o  r  parallel  to  A'  B',  and  through  the  opposite 
extremity,  m,  draw  a  perpendicular,  r  s,  to  this  line,  and  set  oif  on  the  perpen- 
dicular the  distance  r  s  equal  to  the  side  of  the  square,  and  join  s  o.  Now, 
draw  through  the  point  A',  in  the  original  figure,  a  line,  A'  #',  parallel  to  s  o, 
intersecting  the  circle  A'  a'  B'  in  the  point  a',  which,  being  projected  by  a  line 
parallel  to  the  axis  of  the  cylinder, 
and  meeting  the  line  A  a,  drawn  at 
an  angle  of  45°,  gives  the  first  point 
a  in  the  curve  C  d  a.  The  other 
points  will  be  obtained  in  like  man- 
ner, by  drawing  at  pleasure  other 
lines,  such  as  D'  d',  parallel  to  A'  a1. 

To  find  the  outline  of  the  shadow 
cast  into  the  interior  of  a  hollow 
hemisphere  (Fig.  231). 

Let  A  B  0  D  represent  the  hori- 
zontal projection  of  a  concave  hem- 
isphere. Here  it  is  sufficiently  ob- 
vious that,  if  we  draw  through  the 
center  of  the  sphere  a  line  perpen- 
dicular to  the  ray  of  light  A  C,  the 
points  B  and  D  will  at  once  give 
the  extremities  of  the  curves  sought.  On  any  point  of  B  D  produced  as  0', 
construct  the  semicircle  A'  a'  C'  with  a  radius.  A'  0',  equal  to  A  0.  At  A' 
draw  the  line  A'  a',  making  an  angle  of  35°  16'  with  A'  C'.  This  angle,  as  has 
been  said  before,  is  equal  to  that  made  by  the  ray  of  light  in  space  with  the 


FIG.  231. 


120 


SHADES   AND   SHADOWS. 


/ft- 


0' 

r  \ 


planes  of  projection.  The  point  of  intersection  of  the  line  with  the  semicircle 
at  a'  projected  to  a,  gives  a  point  of  the  outline  of  the  shadow.  Similar  sec- 
tions, as  E  F  parallel  to  A  C,  will  give  other  points.  But,  as  this  outline  cover 
of  the  shadow  is  an  ellipse  whose  axes  are  B  D  and  twice  0  a,  it  may  be 
constructed,  when  the  point  a  is  determined,  by  the  ordinary  methods  for 
ellipses. 

To  construct  the  outlines  of  the  shadow  in  the  interior  of  a  concave  sur- 
face, formed  l>y  the  combination  of  a  hollow  semi-cylinder  and  a  quadrant  of 

a  hollow  sphere,  called  a  niche 
(Fig.  232). 

We  already  know  the  mode 
of  tracing  the  shadows  upon 
each  of  these  figures  separately. 
\  \X       \  Thus,  the  shadow  of  the  circu- 

.}••'-.  >- •-••  \         lar  outline  upon  the  spherical 

,£%>-.  >-B    portion  is  part  of  an  ellipse, 

;^->.  _  i  c  D,  found  precisely  as  in  the 

previous  example.  The  point 
e,  where  this  ellipse  cuts  the  horizontal  diam- 
eter A  F,  limits  the  cast  shadow  upon  the  spher- 
ical surface  ;  therefore,  all  the  points  beneath  it 
must  be  determined  upon  the  cylindrical  part. 
Through  A'  in  the  plan  draw  the  line  A'  a'  par- 
allel to  the  ray  of  light ;  project  a'  till  it  inter- 
sects the  line  of  light  A  a  in  the  elevation  at  a. 
The  line  of  shadow  below  a  is  the  shadow  of 
the  edge  of  the  cylinder,  and  must  therefore  be 
a  straight  line.  The  line  of  shadow  between  a 
and  e  is  produced  by  the  outline  of  the  circular 
part  falling  on  a  cylindrical  surface,  and  is  es- 
tablished as  in  previous  constructions. 

To  find  the  line  of  shade  in  a  sphere,  and 
the  outline  of  its  shadow  cast  upon  the  hori- 
zontal plane  (Fig.  233). 

The  line  of  shade  in  a  sphere  is  simply  the 
circumference  of  a  great  circle,  the  plane  of 
which  is  perpendicular  to  the  direction  of  the 

luminous  rays,  and  consequently  inclined  to  the  two  planes  of  projection.  This 
line  will  therefore  be  represented  in  elevation  and  plan  by  two  equal  ellipses, 
the  major  axes  of  which  are  obviously  the  diameters  C  D  and  C'  D',  drawn  at 
an  angle  of  45°. 

To  find  the  minor  axes  of  these  curves,  assume  any  point,  O2,  upon  the  pro- 
longation of  the  diameter  of  the  perpendicular  C'  D',  draw  through  this  point 
the  straight  line  O2  o',  inclined  at  an  angle  of  35°  16',  to  A'  B'  or  its  parallel,  and 
erect  upon  it  the  perpendicular  E3  F2.  The  projection  of  the  two  extremities 
E9  and  F2  upon  the  line  A'  B'  will  give  in  the  plan  the  line  E'  F'  for  the 
length  of  the  required  minor  axis  of  the  ellipse,  i.  e.,  of  the  line  of  shade  in  the 


FIG.  232. 


SHADES   AND   SHADOWS. 


121 


plan  ;  and  this  line,  being  again  transferred  to  the  elevation,  determines  the 
minor  axis  E  F  of  the  line  of  shade  in  the  elevation. 

Supposing  it  were  required  to  draw  these  ellipses,  not  by  means  of  their 
axes,  but  by  points,  any  number  of  these  may  be  obtained  by  making  horizontal 
sections  of  the  sphere.  Thus,  for  example,  if  we  draw  the  chord  G-  H  parallel 
to  A'  B',  to  represent  one  of  these  sections,  and  from  the  point  a,  where  it  cuts 
the  diameter  Ea  F2,  if  we  draw  a  perpendicular  to  A'  B',  the  points  a'  a',  where 


FIG.  233. 

it  intersects  the  circumference  of  the  circle  G'  a'  H',  representing  the  section 
G  H  in  plan,  will  be  two  points  in  the  line  of  shade  required.  These  points 
may  be  transferred,  by  supposing  a  section,  g  h,  to  be  made  in  the  elevation  cor- 
responding to  G-  H,  and  projecting  the  points  a!  a'  by  perpendiculars  to  g  h, 
the  line  representing  the  cutting  plane. 

The  outline  of  the  shadow  cast  by  the  sphere  upon  the  horizontal  plane  is 
also  obviously  an  ellipse  ;  it  may  be  constructed  either  by  means  of  its  two  axes 
or  by  the  help  of  points,  in  the  manner  indicated  in  the  figure. 

To  draw  the  line  of  shade  on  the  surface  of  a  ring  of  circular  section,  in 
vertical  section,  elevation,  and  plan  (Fig.  234). 


122 


SHADES   AND   SHADOWS. 


AVe  shall  first  point  out  the  mode  of  obtaining  those  primary  points  in  the 
curve  which  are  most  easily  found,  and  then  proceed  to  the  general  case  of  de- 
termining any  point  whatever. 
If  tangents  be  drawn  to  the 
circles  represented  in  both  ele- 
vation and  section,  parallel  to 
the  rays  of  light,  their  points 
of  contact,  #,  b,  c,  d,  will  be 
the  starting-points  of  the  re- 
quired lines  of  shade. 

Again,  the  intersections  of 
the  horizontal  lines  ae,dg,  cf, 
drawn  through  these  points, 
with  the  axis  of  the  ring,  will 
give  so  many  new  points,  e,  g, 
f,  in  the  curve.  These  points 
are  denoted  in  the  plan  by 
setting  off  the  distances  a  e 
and  c  f  upon  the  vertical  line 
g'  D,  on  both  sides  of  the  cen- 
ter C'. 

Further,  the  diameter  F' 
G',  drawn  at  an  angle  of  45°, 
determines,  by  its  intersec- 
tions with  the  exterior  and 
interior  circumferences  of  the 
ring,  four  other  points,  F',  t', 
x',  and  G',  in  the  curve  in 
question  ;  these  points  are  all 
to  be  projected  vertically  upon 
the  line  A  B. 

And,  lastly,  to  obtain  the 
lowest  points,  I  ly  draw  tangents 
to  the  circles  in  elevation  and 

section  at  an  angle  of  35°  16'  with  the  ground-line,  and  transfer  the  distances  be- 
tween the  points  of  contact,  s,  s,  and  the  axis  of  the  ring,  to  the  diameter  E'  J', 
where  they  are  denoted  by  I'  I' ;  these  points  are  then  projected  to 
1)  I,  upon  the  horizontal  lines  drawn  through  the  same  points  s,  s. 
To  determine  any  other  points,  draw  through  the  center  C'  a 
diameter,  I'  H',  in  any  direction.  Draw  through  o',  one  of  the 
angular  points  of  the  horizontal  projection  of  the  cube,  made  at 
any  convenient  size  (Fig.  235),  a  straight  line,  o'  r',  parallel  to 
I'  H',  and  from  the  opposite  point  m'  draw  a  perpendicular,  m'  r', 
to  o'  r'.  Then,  having  revolved  the  point  r'  to  ra,  and  projected 
r2  to  r,  join  o  and  r. 

Applying  this  construction  to  the  figures  before  us,  we  now  draw  tangents 
to  the  circles  represented  in  elevation  and  section  parallel  to  the  line  o  r,  and. 


234. 


SHADES  AND   SHADOWS. 


123 


taking  as  radii  the  distances  from  their  respective  points  of  contact,  h  and  I,  to 
the  axis  of  the  ring,  we  describe  corresponding  circles  about  the  center  C'  of 
the  plan.  We  thus  obtain  four  other  points  in  the  curves  required,  namely, 
I',  i',  li' ',  and  H',  which  may  also  be  projected  upon  the  horizontal  lines  drawn 
through  the  points  h  or  I. 

By  drawing  the  straight  line  J'  K'  so  as  to  form  with  F'  G'  the  same  angle 
which  the  latter  makes  with  the  line  H'  I',  we  obtain,  by  the  intersection  of 
that  line  with  the  circles  last  named,  four  other  points  of  the  curves  in 
question. 

To  determine  the  shadows  cast  upon  the  surfaces  of  grooved  pulleys 
(Fig.  236). 

The  construction  of  cast  shadows  upon  surfaces  of  the  kind  now  under  con- 
sideration is  founded  upon  the  principle,  already  announced,  that  when  a  circle 
is  parallel  to  a  plane,  its  shadow,  cast  upon  that  plane,  is  another  circle  equal 
to  the  original  circle. 

Take  the  case  of  a  circular-grooved  pulley  ;  the  cast  shadow  on  its  surface 
is  obviously  derived  from  the  circumference  of  the  upper  edge  A  B.  To  deter- 
mine its  outline,  take  any  horizontal  line 
D  E  in  upper  fig.  and  describe  from  the  cen- 
ter C'  a  circle  with  a  radius  equal  to  the  half 
of  that  line  ;  then  draw  through  the  same 
center  a  line  parallel  to  the  ray  of  light, 
which  will  intersect  the  plane  D  E  in  c ; 
lastly,  describe  from  the  point  c',  as  a  cen- 
ter, an  arc  of  a  circle  with  a  radius  equal  to 
A  C  ;  the  point  of  intersection,  a',  of  this 
arc,  with  the  circumference  of  the  section 
D  E,  will  give,  when  projected'  to  a,  one  of 
the  points  in  the  curve  required. 

To  avoid  unnecessary  labor  in  drawing 
more  lines  parallel  to  D  E  than  are  required, 
it  is  important,  in  the  first  place,  to  ascer- 
tain the  highest  point  in  the  curve  sought. 
This  point  is  the  shadow  of  that  marked  H 
on  the  upper  edge  of  the  pulley,  and  which 
"is  determined  by  the  intersection  of  the  ray 
C'  H'  with  the  circumference  of  that  edge  in  the  plan  ;  and  it  is  obtained  by 
drawing  through  the  point  A  a  straight  line  at  an  angle  of  35°  16'  with  the  line 
A  B,  and  through  the  point  e,  striking  a  horizontal  line  ef,  which  by  its  inter- 
section with  the  line  H  h,  drawn  at  an  angle  of  45°,  will  give  the  point  sought. 

In  the  plan,  the  pulley  is  supposed  to  be  divided  horizontally  in  the  center, 
and  the  shadow  represented  is  derived  from  the  smaller  circle,  and  is  easily 
constructed  by  methods  already  described. 

To  trace  the  outlines  of  the  shadows  cast  upon  the  surfaces  of  a  square- 
threaded  nut  and  screiv  (Figs.  237,  238). 

Fig.  238  represents  the  projections  of  a  screw  with  a  single  square  thread, 
and  placed  in  a  horizontal  position,  A'  a'  being  the  direction  of  the  ray  of  light. 


FIG.  236. 


124 


SHADES  AND  SHADOWS. 


In  this  example,  the  shadow  to  be  determined  is  simply  that  cast  by  the  outer 
edge,  A  B,  of  the  thread  upon  the  surface  of  the  inner  cylinder  ;  therefore,  its 
outline  is  to  be  delineated  in  the  same  manner  as  we  have  already  pointed  out, 
in  treating  of  a  cylinder  surmounting  another  of  smaller  diameter  (Fig.  223). 


A    E'K' 


FIG.  237. 


FIG.  238. 


The  shadow  cast  by  the  helix  ABC  upon  the  concave  surface  of  the  square- 
threaded  nut  is  a  curve,  a  ~b  C  (Fig.  237),  which  is  to  be  determined  in  the  same 
way  as  that  in  the  interior  of  a  hollow  cylinder.  The  same  observation  applies 
to  the  edges  A  A2  and  A8  E,  as  well  as  to  those  of  the  helix  F  G  H  and  the 
edge  H  I.  With  regard  to  the  shadow  of  the  two  edges  J  K  and  K  L,  they 
will  follow  the  rules  laid  down  in  reference  to  the  following  figures,  seeing  that 
they  are  thrown  upon  an  inclined  helical  surface,  of  which  A  L  is  the  gene- 
ratrix. 

To  determine  the  outlines  of  the  shadows  cast  upon  the  surfaces  of  a  triangu- 
lar-threaded nut  and  screw  (Figs.  239,  240). 

Fig.  240  represents  the  case  of  a  triangular-threaded  screw,  and  does  not 
admit  of  so  easy  a  solution  as  the  square-threaded,  because  the  outer  edge, 
A  0  D,  of  the  thread,  in  place  of  throwing  its  shadow  upon  a  cylinder,  pro- 
jects it  upon  a  helical  surface  inclining  to  the  left,  of  which  the  generatrix 
is  known.  Describe  from  the  center  0  a  number  of  circles,  representing  the 
bases  of  so  many  cylinders,  on  the  surfaces  of  which  we  must  suppose  helical 
lines  to  be  traced,  of  the  same  pitch  as  those  which  form  the  exterior  edges 


SHADES  AND   SHADOWS. 


125 


of  the  screw.  We  must  now  draw  any  line,  such  as  B'  E',  parallel  to  the  ray  of 
light,  and  cutting  all  the  circles  described  in  the  plan  in  the  points  B',  F',  GT, 
E',  which  are  then  to  be  successively  projected  to  their  corresponding  helical 
lines  in  the  elevation,  where  they  are  denoted  by  B2,  F,  Gr,  and  E.  Then, 
transferring  the  point  B'  to  its  appropriate  position,  B,  on  the  edge  A  C  D,  and 


FIG.  239. 


FIG.  240. 


drawing  through  the  latter  a  line,  B  #,  at  an  angle  of  45°,  its  intersection  with 
the  curve  B2  G  E  will  give  one  point  in  the  curve  of  the  shadow  required.  In 
the  same  manner,  by  constructing  other  curves,  such  as  H2  J  K,  the  remaining 
points,  as  Ji,  in  the  curve  may  be  found. 

The  same  processes  are  requisite  in  order  to  determine  the  outlines  of  the 
shadows  cast  into  the  interior  surfaces  of  the  corresponding  nut,  as  will  be 
evident  from  an  inspection  of  Fig.  239.  These  shadows  are  derived  not  only 
from  the  helical  edge  A  B  D,  but  also  from  that  of  the  generatrix  A  C. 

The  principles  so  fully  laid  down  and  illustrated  in  the  preceding  pages 
will  be  found  to  admit  of  a  ready  and  simple  application  to  the  delineation  of 
the  shadows  of  all  the  ordinary  forms  and  combinations  of  machinery  and 
architecture,  however  varied  or  complicated  ;  and  the  student  should  exercise 
himself,  at  this  stage  of  his  progress,  in  tracing,  according  to  the  methods 
above  explained,  the  outlines  of  the  cast  shadows  of  pulleys,  spur-wheels,  and 
such  simple  and  elementary  pieces  of  machinery.  It  must  be  observed  that 
the  student  should  never  copy  the  figures  as  here  represented,  but  should  adopt 
some  convenient  scale  somewhat  larger  than  our  figures,  and  construct  his 


126  SHADES  AND   SHADOWS. 

drawings  according  to  the  description,  looking  to  the  figures  as  mere  illustra- 
tions ;  in  this  way,  the  principles  of  the  construction  will  be  more  surely  under- 
stood, and  more  firmly  fixed  in  his  mind. 

MANIPULATION   OF   SHADING   AND    SHADOWS. — METHODS   OF   TINTING. 

The  intensity  of  a  shade  or  shadow  is  regulated  by  the  various  peculiarities 
in  the  forms  of  bodies,  and  by  the  position  which  objects  may  occupy  in  refer- 
ence to  the  light. 

Surfaces  in  the  Light. — Flat  surfaces  wholly  exposed  to  the  light,  and  at 
all  points  equidistant  from  the  eye,  should  receive  a  uniform  tint. 

In  geometrical  drawings,  every  surface  parallel  to  the  plane  of  projection 
is  supposed  to  have  all  its  parts  at  the  same  distance  from  the  eye  ;  such  is 
the  vertical  side  of  the  prism  abed  (Fig.  4,  PL  I).  When  two  surfaces  thus 
situated  are  parallel,  the  one  nearer  the  eye  should  receive  a  lighter  tint  than 
the  other.  Every  surface  exposed  to  the  light,  but  not  parallel  to  the  plane 
of  projection,  and  therefore  having  no  two  points  equally  distant  from  the  eye, 
should  receive  an  unequal  tint.  The  tint  should  gradually  increase  in  depth 
as  the  parts  of  such  a  surface  recede  from  the  eye.  This  effect  is  represented 
in  the  same  figure  on  the  inclined  surface,  a  dfe. 

If  two  surfaces  are  unequally  exposed  to  the  light,  the  one  which  is  more 
nearly  perpendicular  to  the  rays  should  receive  the  fainter  tint. 

Thus,  the  face  e'  a'  (Fig.  1,  PI.  1),  presenting  itself  more  directly  to  the  rays 
of  light  than  the  face  a'  b',  receives  a  tint  which,  although  graduated  in  conse- 
quence of  the  inclination  of  this  face  to  the  plane  of  projection,  becomes  at 
that  part  of  the  surface  situated  nearest  to  the  eye  fainter  than  the  tint  on  the 
surface  a'  b' . 

/Surfaces  in  Shade. — When  a  surface  entirely  in  the  shade  is  parallel  to  the 
plane  of  projection,  it  should  receive  a  uniform  dark  tint. 

When  two  objects  parallel  to  each  other  are  in  the  shade,  the  one  nearer 
the  eye  should  receive  the  darker  tint. 

When  a  surface  in  the  shade  is  inclined  to  the  plane  of  projection,  those 
parts  which  are  nearest  to  the  eye  should  receive  the  deepest  tint.  This  can  be 
seen  on  the  face  bg  h  c  (Fig.  4),  where  the  tint  is  much  darker  toward  the  line 
b  c,  than  where  it  approaches  the  line  g  h. 

If  two  surfaces  exposed  to  the  light,  but  unequally  inclined  to  its  rays,  have 
a  shadow  cast  upon  them,  that  part  of  it  which  falls  upon  the  surface  more 
directly  influenced  by  the  light  should  be  darker  than  where  it  falls  upon  the 
other  surface. 

Exemplifications  of  the  foregoing  rules  may  be  seen  on  the  various  figures 
of  Plates  I  to  V. 

In  order  that  these  rules  may  be  practiced  with  proper  effect,  we  shall  give 
some  directions  for  using  the  brush  or  hair-pencil,  and  explain  the  usual  meth- 
ods employed  for  tinting  and  shading. 

The  methods  of  shading  most  generally  adopted  are  either  by  the  superpo- 
sition of  any  number  of  flat  tints,  or  by  tints  softened  off  at  their  edges.  The 
former  method  is  the  more  simple  of  the  two,  and  should  be  the  first  attempted. 


) 

SHADES  A^D  SHADOWS.  127 

To  shade  a  prism  by  means  of  flat  tints  (PL  I). 

According  to  the  position  of  the  prism,  as  shown  by  its  plan,  the  face  abed 
(Fig.  4)  being  parallel  to  the  plane  of  projection,  should  receive  a  uniform  tint 
either  of  India  ink  or  sepia.  When  the  surface  to  be  tinted  happens  to  be 
very  large,  it  is  advisable  to  put  on  a  very  light  tint  first,  and  then  to  go  over 
the  surface  a  second  time  with  a  tint  sufficiently  dark  to  give  the  desired  tone 
to  the  surface. 

The  face  b  g  h  c  being  inclined  to  the  plane  of  projection,  should  receive  a 
graduated  tint  from  the  line  b  e  to  the  line  g  li.  This  gradation  is  obtained  by 
laying  on  a  succession  of  flat  tints  in  the  following  manner  :  First,  divide  the 
plan  b'  g'  into  equal  parts  at  the  points  1',  2',  and  from  these  points  project 
lines  upon,  and  parallel  to,  the  sides  of  the  face  b  g  h  c.  These  lines  should  be 
drawn  very  lightly  in  pencil,  as  they  merely  serve  to  circumscribe  the  tints. 
A  grayish  tint  is  then  spread  over  that  portion  of  the  face  b  g  h  c  (Fig.  2), 
between  the  lines  b  c  and  1,  1.  When  this  is  dry,  a  similar  tint  is  to  be  laid 
on,  extending  over  the  space  comprised  within  the  lines  b  c  and  2,  2  (Fig.  3). 
Lastly,  a  third  tint,  covering  the  whole  surface  bclig  (Fig.  4)  imparts  the 
desired  graduated  shade  to  that  side  of  the  prism.  The  number  of  tints 
designed  to  express  such  a  graduated  shade  depends  upon  the  size  of  the  sur- 
face to  be  shaded ;  and  the  depth  of  tint  must  vary  according  to  this  number. 

As  the  number  of  these  washes  is  increased,  the  whole  shade  gradually  pre- 
sents a  softer  appearance,  and  the  lines  which  border  the  different  tints  become 
less  harsh  and  perceptible.  For  this  reason  the  foregoing  method  of  represent- 
ing a  shade  or  graduated  tint  by  washes  successively  passing  over  each  other 
is  preferable  to  that  sometimes  employed,  of  first  covering  the  whole  surface 
bg  he  with  a  faint  tint,  then  putting  on  a  second  tint  b  2  2  c,  followed,  lastly, 
by  a  narrow  wash  b  1  1  c  ;  because,  in  following  this  process,  the  outline  of  each 
wash  remains  untouched,  and  presents,  unavoidably,  a  prominence  and  harsh- 
ness which,  by  the  former  method,  are  in  a  great  measure  subdued. 

The  face  a  dfe  being  also  inclined  to  the  plane  of  projection,  should,  as  it 
is  entirely  in  the  light,  be  covered  by  a  series  of  much  fainter  tints  than  the 
surface  bghc,  which  is  in  the  shade,  darkening,  however,  toward  the  line  ef. 
The  gradation  of  tint  is  effected  in  the  same  way  as  on  the  face  b  g  h  c. 

To  shade  a  cylinder  by  means  of  flat  tints  (PL  I). 

In  shading  a  cylinder,  it  will  be  necessary  to  consider  the  difference  in  the 
tone  proper  to  be  maintained  between  the  part  in  the  light  and  that  in  the  shade. 
It  should  be  remembered  that  the  line  of  separation  between  the  light  and 
shade  a  b  (Fig.  6)  is  determined  by  the  radius  0  a'  (Fig.  5),  drawn  perpendicu- 
lar to  the  rays  of  light  E  0.  That  part,  therefore,  of  the  elevation  of  the  cylin- 
der which  is  in  the  shade  is  comprised  between  the  lines  a  b  and  c  d.  This 
portion,  then,  should  be  shaded  conformably  to  the  rule  previously  laid  down 
for  treating-  surfaces  in  the  shade  inclined  to  the  plane  of  projection.  All  the 
remaining  part  of  the  cylinder  which  is  visible  presents  itself  to  the  light ;  but, 
in  consequence  of  its  circular  figure,  the  rays  of  light  form  angles  varying  at 
every  part  of  its  surface,  and  consequently  this  surface  should  receive  a  gradu- 
ated tint.  In  order  to  represent  with  effect  the  rotundity,  it  will  be  neces- 
sary to  determine  with  precision  the  part  of  the  surface  which  is  most  directly 


128  SHADES  AND   SHADOWS. 

affected  by  the  light.  This  part,  then,  is  situated  about  the  line  e  i  (Fig.  12), 
As  the  visual  rays,  however,  are  perpendicular  to  the  vertical  plane,  and  there- 
fore parallel  to  V  0,  it  follows  that  the  part  which  appears  clearest  to  the  eye 
will  be  near  this  line  V  0,  and  may  be  limited  by  the  line  T  0,  which  bisects 
the  angle  V  0  K.  By  projecting  the  points  e'  and  m',  and  drawing  the  lines  e  i 
and  m  n  (Fig.  12),  the  surface  comprised  between  these  lines  will  represent  the 
lightest  part  of  the  cylinder. 

This  part  should  have  no  tint  upon  it  whatever  if  the  cylinder  happen  to 
be  polished — a  turned  iron  shaft  or  a  marble  column,  for  instance  ;  but  if  the 
surface  of  the  cylinder  be  rough,  as  in  the  case  of  a  cast-iron  pipe,  then  a  very 
light  tint — considerably  lighter  than  on  any  other  part — may  be  given  it. 

Again,  let  us  suppose  the  half-plan  of  the  cylinder  /'  m'  a'  c'  to  be  divided 
into  any  number  of  equal  parts.  Indicate  these  divisions  upon  the  surface  of 
the  cylinder  by  faint  pencil-lines,  and  begin  the  shading  by  laying  a  tint  over 
all  that  part  of  the  cylinder  in  the  shade  a  c  d  b  (Fig.  6).  This  will  at  once 
render  evident  the  light  and  dark  parts  of  the  cylinder.  When  this  is  dry, 
put  on  a  second  tint  covering  the  line  a  1)  of  separation  of  light  and  shade,  and 
extending  over  one  division,  as  shown  in  Fig.  7.  Proceed  in  this  way  until 
the  whole  of  that  part  of  the  cylinder  which  is  in  the  shade  is  covered.  The 
successive  stages  of  this  process  may  be  seen  in  Figs.  6  to  12. 

Treat  in  a  similar  manner  the  part/e  ig  (Fig.  12),  and  complete  the  opera- 
tion by  covering  the  whole  surface  of  the  cylinder — excepting  only  the  division 
e  m  n  i — with  a  very  light  tint ;  the  cylinder  will  then  assume  the  appearance 
presented  by  Fig.  12. 

To  shade  the  segment  of  an  hexagonal  pyramid  by  means  of  softened  tints- 

(PI.  n). 

The  plan  of  this  figure  is  similar  to  that  of  the  prism  (PI.  I).  Its  position 
in  reference  to  the  light  is  also  the  same.  Thus,  the  face  abed  should  receive 
a  uniform  flat  tint.  If,  however,  it  be  desired  to  adhere  rigorously  to  the  pre- 
ceding rules,  the  tint  may  be  slightly  deepened  as  it  approaches  the  top  of  the 
pyramid,  seeing  that  the  surface  is  not  quite  parallel  to  the  vertical  plane. 

The  face  b  g  li  c  being  inclined  and  in  the  shade,  should  receive  a  dark  tint. 
The  darkest  part  of  this  tint  is  where  it  meets  the  line  b  c,  and  gradually  be- 
comes lighter  as  it  approaches  the  line  g  h.  To  produce  this  effect,  apply  a 
narrow  strip  of  tint  to  the  side  b  c  (Fig.  6),  and  then,  qualifying  the  tint  in 
the  brush  with  a  little  water,  join  another  strip  to  this,  and  finally,  by  means 
of  another  brush  moistened  with  water,  soften  off  this  second  strip  toward  the 
line  1,  1,  which  may  be  taken  as  the  limit  of  the  first  tint. 

When  the  first  tint  is  dry,  cover  it  with  a  second,  which  must  be  similarly 
treated,  and  should  extend  beyond  the  first  up  to  the  line  2,  2  (Fig.  7).  Pro- 
ceed in  this  manner  with  other  tints,  until  the  whole  face  bg he  is  shaded,  as 
presented  in  Fig.  8. 

In  the  same  way  the  face  e  a  d  f  is  to  be  covered,  though  with  a  consid- 
erably lighter  tint,  for  the  rays  of  light  happen  to  fall  upon  it  almost  perpen- 
dicularly. 

It  may  be  observed  that,  consistently  to  carry  out  the  rules  we  have  laid 
down,  the  tint  on  these  two  faces  should  be  slightly  graduated  from  eatofd, 


SHADES  AND  SHADOWS.  129 

and  from  c  li  to  b  g.  But  this  exactitude  may  be  disregarded  until  some  pro- 
ficiency in  shading  has  been  acquired. 

To  shade  a  cylinder  ~by  means  of  softened  tints  (PL  II). 

The  boundary  of  each  tint  being  indicated  in  a  manner  precisely  similar  to 
that  shown  in  PL  I,  the  first  strip  of  tint  must  cover  the  line  of  extreme  shade 
a  I,  and  then  be  softened  off  on  each  side.  Other  and  successively  wider  strips 
of  tint  are  to  follow,  and  receive  the  same  treatment  as  the  one  first  put  on. 
The  results  of  this  process  are  shown  in  the  figures. 

As  this  method  requires  considerable  practice  before  it  can  be  performed 
with  much  nicety,  the  learner  need  not  be  discouraged  at  the  failure  of  his  first 
attempts,  but  persevere  in  practicing  on  simple  figures  of  different  sizes. 

If,  after  shading  a  figure  by  the  foregoing  method,  any  very  apparent  ine- 
qualities present  themselves  in  the  shades,  such  defects  may  be  remedied,  in 
some  measure,  by  washing  off  excesses  of  tint  with  a  brush  or  a  damp  sponge, 
and  by  supplying  a  little  color  to  those  parts  which  are  too  light. 

Dexterity  in  shading  figures  by  softened  tints  will  be  facilitated  in  prac- 
ticing upon  large  surfaces  ;  this  will  be  the  surest  way  of  overcoming  that 
timidity  and  hesitation  which  usually  accompany  all  first  attempts,  but  which 
must  be  laid  aside  before  much  proficiency  in  shading  can  be  acquired. 


ELABORATION   OF   SHADING   AND   SHADOWS. 

Thus  far  the  simplest  primary  rules  for  shading  isolated  objects  have  been 
laid  down,  and  the  easiest  methods  of  carrying  them  into  operation  explained. 
It  is  now  proposed  to  exemplify  these  rules  upon  more  complex  forms,  to  show 
where  the  shading  may  be  modified  or  exaggerated,  to  introduce  additional 
rules  more  especially  adapted  for  mechanical  coloring,  and  to  offer  some  obser- 
vations and  directions  for  effectively  shading  the  drawing  of  machines  in  their 
entity. 

Whatman's  best  rough-grained  drawing-paper  is  better  adapted  for  receiving- 
color  than  any  other.  Of  this  paper,  the  double-elephant  size  is  preferable,  as 
it  possesses  a  peculiar  consistency  and  grain.  The  face  of  the  paper  to  be  used 
is  the  one  on  which  the  water-mark  is  read  correctly. 

The  paper  for  a  colored  drawing  ought  always  to  be  strained  upon  a  board 
with  glue  or  strong  gum.  Before  doing  this,  care  must  be  taken  to  dampen  the 
face  of  the  paper  with  a  sponge  well  charged  with  water,  in  order  to  remove 
any  impurities  from  its  surface,  and  as  a  necessary  preparation  for  the  better 
reception  of  the  color.  The  sponge  should  merely  touch  the  paper  lightly,  and 
not  rub  it.  The  whole  of  the  surface  is  to  be  dampened,  that  the  paper  may 
be  subjected  to  a  uniform  degree  of  expansion,  thereby  insuring,  as  it  dries, 
a  uniform  degree  of  contraction.  Submitted  to  this  treatment,  the  sheet  of 
paper  will  present,  when  thoroughly  dry,  a  clean,  smooth  surface,  agreeable 
to  work  upon . 

The  size  of  the  brushes  to  be  used  will,  of  course,  depend  upon  the  scale  to 
which  the  drawing  is  made.     Long,  thin  brushes,  however,  should  be  avoided. 
Those  possessing  corpulent  bodies  and  fine  points  are  to  be  preferred,  as  they 
retain  a  greater  quantity  of  color,  and  are  more  manageable. 
9 


130  SHADES  AND  SHADOWS. 

During  the  process  of  laying  on  a  flat  tint,  if  the  surface  be  large — though 
this  is  seldom  the  case  except  in  topographical  drawings — the  drawing  may  be 
slightly  inclined,  and  the  brush  well  charged  with  color,  so  that  the  edge  of 
the  tint  may  be  kept  in  a  moist  state  until  the  whole  surface  is  covered.  In 
tinting  a  small  surface,  the  brush  should  never  have  much  color  in  it,  for,  if  it 
have,  the  surface  will  unavoidably  present  coarse,  rugged  edges,  and  a  coarse, 
uneven  appearance  throughout. 

In  the  examples  of  shading  which  are  given  in  this  work,  it  may  be  observed 
that  all  objects  with  curved  outlines  have  a  certain  amount  of  reflected  light 
imparted  to  them.  It  is  true  that  all  bodies,  whatever  may  be  their  form,  are 
aifected  by  reflected  light ;  but,  with  a  few  exceptions,  this  light  is  only  appre- 
ciable on  curved  surfaces. 

All  bodies  in  the  light  reflect  on  those  objects  which  surround  them  more 
or  less  light  according  to  the  situation.  Wherever  light  extends,  reflection 
follows.  If  an  object  be  isolated,  it  is  still  reached,  by  reflected  light,  from 
the  ground  on  which  it  rests,  or  from  the  air  which  surrounds  it. 

In  proportion  to  the  degree  of  polish  or  brightness  in  the  color  of  a  body,  is 
the  amount  of  reflected  light  which  it  spreads  over  adjacent  objects,  and  also 
its  own  susceptibility  of  illumination  under  the  reflection  from  other  bodies. 
A  polished  steam-cylinder  or  a  white  porcelain  vase  receives  and  imparts  more 
reflected  light  than  a  rough  casting  or  a  stone  pitcher. 

Shade,  even  the  most  inconsiderable,  ought  never  to  extend  to  the  outline 
of  any  smooth  circular  body.  On  a  polished  sphere,  for  instance,  the  shade 
should  be  delicately  softened  off  just  before  it  meets  the  circumference,  and, 
when  the  shading  is  completed,  the  body-color  intended  for  the  sphere  may  be 
carried  on  to  its  outline.  This  will  give  a  transparency  to  that  part  of  the 
sphere  influenced  by  reflected  light,  which  it  could  not  have  possessed  if  the 
shade-tint  had  been  extended  to  its  circumference.  Very  little  shade  should 
be  suffered  to  reach  the  outlines  even  of  rough  circular  bodies,  lest  the  coloring 
look  harsh,  and  present  a  coarse  appearance  quite  at  variance  with  its  natural 
aspect.  Shadows  also  become  lighter  as  they  recede  from  the  bodies  which  cast 
them,  owing  to  the  increasing  amount  of  reflection  which  falls  on  them  from 
surrounding  objects. 

Shadotvs  appear  to  increase  in  depth  as  their  distance  from  the  spectator 
diminishes.  In  nature  this  increase  is  only  appreciable  at  considerable  dis- 
tances. Even  on  extensive  buildings,  inequalities  in  the  depth  of  the  shadows 
are  hardly  perceptible  ;  much  less,  then,  can  any  natural  gradation  present 
itself  in  the  shadows  on  a  machine,  which,  supposing  it  to  be  of  the  largest 
construction,  is  confined  to  a  comparatively  small  space.  It  is  most  important, 
however,  for  the  effective  representation  of  machinery,  that  the  variation  in 
the  distance  of  each  part  of  a  machine  from  the  spectator  should  at  once  strike 
the  eye  ;  and  an  exaggeration  in  expressing  the  varying  depths  of  the  shadows 
is  one  means  of  effecting  that  object.  The  shadows  on  the  nearest  and  most 
prominent  parts  of  a  machine  should  be  made  as  dark  as  color  can  render  them, 
the  colorist  being  thus  enabled  to  exhibit  a  marked  difference  in  the  shadows 
on  the  other  parts  of  the  machine  as  they  recede  from  the  eye.  The  same 
direction  is  applicable  in  reference  to  shades.  The  shade  on  a  cylinder,  for 


SHADES  AND  SHADOWS.  131 

instance,  situated  near  the  spectator,  ought  to  be  darker  than  on  one  more 
remote  ;  in  fact,  the  gradation  of  depth  for  the  shades  follows  that  which  de- 
picts the  shadows.  As  a  general  rule,  the  color  on  a  machine,  no  matter  what 
it  may  be  intended  to  represent,  should  become  lighter  as  the  parts  on  which  it 
is  placed  recede  from  the  eye. 

Plates  III  and  IV  present  some  very  good  examples  of  finished  shading. 

Plate  III  represents,  both  in  elevation  and  plan,  different  solids  variously 
penetrated  and  intersected.  The  rules  for  the  projection  of  these  solids  have 
been  given  under  the  head  of  Orthographic  Projection.  They  are  selected  with 
a  view  of  exhibiting  those  cases  which  are  of  most  frequent  occurrence,  and  at 
the  same  time  elucidating  the  general  principles  of  shading. 

Plate  IV  presents  examples  of  shading  and  shadow. 

Fig.  1  presents  a  hexagonal  prism  surmounted  by  a  fillet.  The  most  notice- 
able part  of  this  figure  is  the  shadow  of  the  prism  in  the  plan  view.  It  presents 
a  good  example  of  the  graduated  expression  which  should  be  given  to  all  shad- 
ows cast  upon  plain  surfaces.  Its  two  extremities  are  remarkably  different  in 
their  tone.  As  the  shadow  nears  the  prism,  it  increases  rapidly  in  depth  ;  on 
the  contrary,  as  it  approaches  the  other  end,  it  assumes  a  comparatively  light 
appearance.  This  difference  is  doubtlessly  a  great  exaggeration  upon  what  it 
would  naturally  display.  Any  modification  of  it,  however,  in  the  representa- 
tion would  destroy  the  best  effect  of  the  shadow. 

The  direction  which  the  shades  and  shadows  take,  in  all  the  plans  of  the 
figures  in  this  plate,  is  from  the  left-hand  lower  corner.  This  is  rigorously 
correct,  supposing  the  objects  to  remain  stationary,  while  the  spectator  views 
them  in  both  a  vertical  and  a  horizontal  position.  Nevertheless,  to  many,  this 
upward  direction  given  to  the  shadows  has  an  awkward  appearance,  and,  per- 
haps, in  the  plan  of  an  entire  machine,  the  shadows  may  look  better  if  their 
direction  coincide  with  that  which  is  given  to  them  in  the  elevation.  If,  how- 
ever, the  shadows  be  correctly  projected,  their  direction  is  an  arbitrary  matter, 
and  may  be  left  to  the  taste  of  the  draughtsman. 

Figs.  2,  3,  and  6  exemplify  the  complex  appearance  of  shade  and  shadow 
presented  on  concave  surfaces.  It  is  worthy  of  particular  notice  that  the 
shadow  on  a  concave  surface  is  darkest  toward  its  outline,  and  becomes  lighter 
as  it  nears  the  edge  of  the  object.  Reflection  from  that  part  of  the  surface  on 
which  the  light  falls  most  powerfully  causes  this  gradual  diminution  in  the 
depth  of  the  shadow,  the  greatest  amount  of  reflection  being  opposite  the  great- 
est amount  of  light. 

It  may  be  as  well  to  remark  here  that  no  'brilliant  or  extreme  light  should 
be  left  on  concave  surfaces,  as  such  lights  would  tend  to  render  it  doubtful  at 
first  sight  whether  the  objects  represented  were  concave  or  convex.  After  the 
body-color — which  shall  be  treated  in  a  subsequent  section — has  been  put  on,  a 
faint  wash  should  be  passed  very  lightly  over  the  whole  concavity.  This  will 
not  only  modify  and  subdue  the  light,  but  tend  to  soften  any  asperities  in  the 
tinting,  which  are  more  unsightly  on  a  concave  surface  than  on  any  other. 

The  lightest  part  of  a  sphere  (Fig.  4)  is  confined  to  a  mere  point,  around 
which  the  shade  commences  and  gradually  increases  as  it  recedes.  This  point 
is  not  indicated  on  the  figure  referred  to,  because  the  shade-tint  on  a  sphere 


132  SHADES  AND  SHADOWS. 

ought  not  to  be  spread  over  a  greater  portion  of  its  surface  than  is  shown  there. 
The  very  delicate  and  hardly  perceptible  progression  of  the  shade  in  the  imme- 
diate vicinity  of  the  light-point  should  be  effected  by  means  of  the  body-color 
of  the  sphere.  If,  for  instance,  the  material  of  which  the  sphere  is  composed 
be  brass,  the  body-color  itself  should  be  lightened  as  it  nears  the  light  point. 
In  like  manner  all  polished  or  light-colored  curved  surfaces  should  be  treated  ; 
the  part  bordering  upon  the  extreme  light  being  covered  with  a  tint  of  body- 
color  somewhat  fainter  than  that  used  for  the  flat  surfaces.  Again,  if  the 
sphere  be  of  cast-iron,  then  the  ordinary  body-color  should  be  deepened  from 
the  light  point  until  it  meets  the  shade-tint,  over  which  it  is  to  be  spread  uni- 
formly. Any  curved  unpolished  surface  is  to  be  thus  treated  ;  the  body-color 
should  be  gradually  deepened  as  it  recedes  from  that  part  of  the  surface  most 
exposed  to  the  light.  Considerable  management  is  necessary  in  order  to  shade 
a  sphere  effectively,  ^he  best  way  is  to  put  on  two  or  three  softened-off  tints 
in  the  form  of  crescencs  converging  toward  the  light-point,  the  first  one  being^ 
carried  over  the  point  of  deepest  shade. 

A  ring  (Fig.  5)  is  a  difficult  object  to  shade.  To  change  with  accurate  and 
effective  gradation  the  shade  from  the  inside  to  the  outside  of  the  ring,  to  leave 
with  regularity  a  line  of  light  upon  its  surface,  and  to  project  its  shadow  with 
precision,  require  a  degree  of  attention  and  care  in  their  execution  greater, 
perhaps,  than  the  shade  and  shadow  of  any  other  simple  figure.  The  learner, 
therefore,  should  practice  the  shading  of  this  figure,  as  he  will  seldom  meet 
with  one  presenting  greater  difficulties. 

Figs.  7  and  8  show  the  peculiarities  of  the  shadows  cast  by  a  conical  form 
on  a  sphere  or  cylinder.  The  following  fact  should  be  well  noted  in  the  mem- 
ory :  That  the  depth  of  a  shadow  on  any  object  is  in  proportion  to  the  degree 
of  light  which  it  encounters  on  the  surface  of  that  object.  In  these  figures 
very  apt  illustrations  of  this  fact  may  be  remarked.  It  will  be  seen,  by  refer- 
ring to  the  plan  (Fig.  7),  that  the  shadow  of  the  apex  of  the  cone  happens  to 
fall  upon  the  lightest  point  of  the  sphere,  and  is,  therefore,  the  darkest  part  of 
the  shadow.  So  also  the  deepest  portion  of  the  shadow  of  the  cone  on  the 
cylinder  in  the  plan  (Fig.  8)  is  exactly  where  it  coincides  with  the  line  of  ex- 
treme light.  Flat  surfaces  are  similarly  affected,  the  shadows  thrown  on  them 
being  less  darkly  expressed,  according  as  their  inclination  to  the  plane  of  pro- 
jection increases.  The  body-color  on  a  flat  surface  should,  on  the  contrary, 
increase  in  depth  as  the  surface  becomes  more  inclined  to  this  plane. 

Another  notable  fact  is  exemplified  by  these  figures — that  reflected  light  is 
incident  to  shadows  as  well  as  to  shades.  This  is  very  observable  where  the 
shadow  of  the  cone  falls  upon  the  cylinder.  It  may  likewise  be  remarked, 
though  to  a  less  extent,  on  other  parts  of  these  figures.  The  reflected  light 
on  the  cone  from  the  sphere  or  cylinder  is  also  worthy  of  observation.  This 
light  adds  greatly  to  the  effect  of  the  shadows,  and,  indeed,  to  the  appearance 
of  the  objects  themselves.  Altogether,  these  figures  offer  admirable  scope  for 
study  and  practice. 

The  concentration  within  a  small  space  of  nearly  all  the  peculiarities  and 
effects  of  light,  shade,  and  shadow,  may  be  seen  on  Plate  V  in  the  examples, 
of  screws  there  given. 


SHADES  AND   SHADOWS.  133 

Under  the  head  of  Topographical,  Mechanical,,  and  Architectural  Drawing, 
will  be  given  examples  of  drawings  in  shade  and  shadow,  and  in  varied  colors 
expressing  conditions  of  surfaces  or  materials  of  composition.  In  the  topo- 
graphical and  architectural  examples,  often  a  certain  amount  of  artistic  effect 
can  be  introduced,  but,  in  the  mechanical,  distinctness  of  outline  and  accuracy 
of  expression  are  essential ;  but,  to  maintain  harmony  in  the  coloring,  and  to 
equalize  the  appearance  of  the  drawing,  large  shades  should  be  colored  less 
darkly  than  small,  as  they  may  be  situated  at  the  same  distance  from  the  eye, 
and  no  very  dark  shading  is  permissible. 

In  preparing  colors  for  tints,  great  care  should  be  used  in  grinding.  The 
end  of  the  cake  should  be  slightly  wetted  and  rubbed  on  a  porcelain  palette, 
and  then  transferred  by  a  wet  brush  to  another  saucer,  and  water  added  to 
bring  to  the  required  tint.  Mixed  colors  should  be  intimately  blended  by  the 
brush.  Grind  in  excess  enough  of  all  the  tint  required,  and  let  it  stand  in  the 
saucer  till  the  grosser  particles  have  settled  and  the  liquid  is  of  clear  and  uni- 
form tint.  It  is  very  common  to  make  little  boxes  or  bag-like  receptacles  of 
waste  drawing-paper  to  hold  the  colors  instead  of  saucers ;  the  gross  matters, 
settling  on  the  bottom,  are  not  then  so  readily  disturbed. 

Instead  of  hard  cakes  of  color,  moist  colors  are  used,  either  in  cakes  or 
collapsible  tubes,  which  preclude  the  necessity  of  grinding.  For  flat  tints  or 
washes,  aniline  colors,  dissolved  in  water  and  kept  bottled,  afford  the  readiest 
means  of  coloring,  but  are  not  applicable  to  finished  work. 

Sometimes  the  surface  of  the  paper  is,  as  it  were,  greasy,  and  resists  colors ; 
in  that  case,  dissolve  a  piece  of  ox-gall,  the  size  of  a  pea,  in  a  tumbler  of 
water,  and  use  this  solution  with  the  colors  instead  of  plain  water. 

When  the  brush  is  too  full,  as  it  comes  toward  the  limit  of  the  tint,  take  up 
the  surplus  moisture  on  a  wet  sponge  or  piece  of  cloth  or  blotting-paper. 

An  expeditious  way  of  shading  a  cylinder  or  expressing  the  shores  of  a 
stream  or  lake,  is  by  drawing  with  a  brush  full  of  the  darkest  tint  along  the 
sides  of  cylinder  or  shores  of  water,  and  then,  with  a  wet  brush,  modifying 
this  tint  toward  the  light  from  the  sides,  so  as  to  give  a  shaded  appearance. 
For  this  purpose,  two  brushes  will  be  necessary,  one  with  color,  the  other  with 
water  ;  also,  a  tumbler  of  water,  and  a  piece  of  blotting-paper,  to  take  up  the 
excess  of  moisture  from  paper  or  brush.  Often  a  single  line  of  dark  color 
blended  this  way  will  express  all  that  is  necessary,  but  the  effect  may  be  im- 
proved by  a  sort  of  stippling  with  the  color-brush  and  extending  the  line  of  shade. 

The  same  effect  is  obtained  better  by  drawing  two  faint  pencil-lines  on  the 
elevation  of  the  cylinder,  for  instance,  to  indicate  the  extremes  of  light  and 
shade  on  its  surface.  Pass  the  brush,  moderately  full  of  the  darkest  tint,  down 
the  line  of  deepest  shade,  spreading  the  color  more  or  less  on  either  side,  accord- 
ing to  the  diameter  of  the  cylinder  ;  then,  if  possible,  before  this  layer  of  tint 
is  dry,  toward  the  line  of  extreme  light,  beginning  at  the  top,  and  encroaching 
slightly  over  the  edge  of  the  first  tint,  lay  on  another  not  quite  so  dark,  but 
about  double  its  width.  It  may  be  observed  that  it  is  not  very  essential  to  put 
on  the  second  tint  befor  the  first  is  dry,  for  the  latter  should  be  so  dark  and 
thick  that  its  edges  may  be  easily  softened  at  any  time.  While  this  second  tint 
is  still  wet,  with  a  much  lighter  color  in  the  brush,  proceed  in  the  same  man- 


134:  SHADES  AND   SHADOWS. 

ner  with  a  third  tint,  and  so  on  until  the  line  of  extreme  light  is  nearly 
attained.  Repeat  this  process  on  the  other  side  of  the  first  tint,  approaching 
the  outline  of  the  cylinder  with  a  very  faint  wash,  so  as  to  represent  the  re- 
flected light  which  progressively  modifies  the  shade  as  it  nears  that  line.  Then 
let  a  darkish  narrow  strip  of  tint  meet  and  pass  along  the  outline  of  the  cylin- 
der on  the  other  side  of  the  extreme  line  of  light,  after  which  gradually  fainter 
tints  should  follow,  treated  in  a  manner  similar  to  that  which  has  been  already 
described,  and  becoming  almost  imperceptible  just  before  arriving  at  the  line 
of  light. 

This  is  a  very  expeditious  way  of  shading  a  cylinder  ;  but  even  to  the  most 
experienced  colorist  it  is  not  possible,  by  the  above-described  means  alone,  to 
impart  a  sufficient  degree  of  well-regulated  rotundity  to  the  appearance  of  such 
an  object.  Superfluities  and  deficiencies  of  color  will  appear  here  and  there. 
It  will  be  necessary,  therefore,  to  equalize  to  some  extent,  by  a  species  of  gross 
stippling,  the  disparities  which  present  themselves.  This  is  done  by  spreading 
a  little  color  over  the  parts  where  it  is  deficient,  and  then  passing  very  lightly 
over  nearly  the  whole  width  of  the  shade,  with  the  brush  supplied  with  a  very 
light  wash.  This  process  may  be  repeated  to  suit  the  degree  of  finish  which  it 
is  desired  to  give  the  drawing.  In  the  same  manner  the  shading  of  all  curved 
surfaces  is  to  be  treated. 

The  shades  being  put  in,  that  of  the  shadows  follows.  The  outline  of  any 
shadow  being  drawn  in  pencil,  along  its  inner  line — the  line  which  forms  a 
portion  of  the  figure  of  the  object  whose  shadow  is  to  be  represented — lay  on  a 
strip  of  the  darkest  tint,  wide  or  narrow,  according  to  the  width  of  the  shadow, 
and  then,  before  it  is  dry,  soften  off  its  outer  edge.  This  may  be  repeated  as 
often  as  the  taste  of  the  colorist  may  dictate,  but  the  color  should  not  spread 
itself  over  much  more  than  half  the  space  occupied  by  the  shadow.  These 
preliminary  touches  will  add  to  the  intensity  of  the  proposed  shadow,  and 
neutralize  a  certain  harshness  of  appearance  inevitable  to  all  shadows  made 
equally  dark  throughout. 

The  finish  is  made  by  a  light  wash  or  two  of  the  body-color,  and  in  passing 
over  the  shades  and  shadows  care  must  be  taken  to  maneuver  the  brush  at 
such  parts  quickly  and  lightly. 

The  shades  and  shadows  of  a  machine  are  modified  in  intensity  as  their 
distance  from  the  eye  increases.  Its  body-color  should  be  treated  in  a  similar 
manner,  becoming  lighter  and  less  bright  as  the  parts  of  the  machine  which  it 
covers  recede  from  the  spectator. 

When  the  large  circular  members  of  a  machine  have  been  shaded,  the  shad- 
ows, and  even  the  body-color  on  those  parts  farthest  removed  from  the  eye,  are 
to  follow,  and  the  proportion  of  India  ink  in  the  tint  used  should  increase  as 
the  part  to  be  colored  becomes  more  remote.  A  little  washing,  moreover,  of 
the  most  distant  parts  is  allowable,  as  it  gives  a  pleasing  appearance  of  atmos- 
pheric remoteness,  or  depth,  to  the  color  thus  treated. 

The  amount  of  light  and  reflection  on  the  members  of  a  machine  should 
diminish  in  intensity  as  the  distance  of  such  objects  from  the  spectator  in- 
creases. As  it  is  necessary,  for  effect,  to  render,  on  those  parts  of  a  machine 
nearest  the  eye,  the  contrast  of  light  and  shade  as  intense  as  possible,  so,  for 


SHADES   AND   SHADOWS.  135 

the  same  object,  the  light  and  shade  on  the  remotest  parts  should  be  subdued 
and  blended  according  to  the  extent  or  size  of  the  machine. 

A  means  of  adding  considerably  to  the  definiteness  of  a  colored  mechanical 
drawing,  and  of  promoting,  in  a  remarkable  degree,  its  effective  appearance, 
is  obtained  by  leaving  a  very  narrow  margin  of  light  on  the  edges  of  all  sur- 
faces, no  matter  what  may  be  the  angles  which  they  may  form  with  the  sur- 
faces that  join  them.  This  should  be  done  invariably  ;  but  the  margin  of 
those  edges  which  happen  to  have  shadows  falling  on  them,  instead  of  being 
left  quite  white,  may  be  slightly  subdued. 

To  effect  this,  suppose  the  object  about  to  receive  the  color  to  be  the  eleva- 
tion of  a  long,  flat  rod  or  lever,  on  the  edge  of  which  a  line  of  light  is  to  be 
left.  Fill  the  drawing-pen  as  full  as  it  will  conveniently  hold  with  tint  des- 
tined to  cover  the  rod  or  lever,  and  draw  a  broad  line  just  within,  but  not 
touching,  the  edge  of  the  lever  exposed  to  the  light.  As  it  is  essential  for  the 
successful  accomplishment  of  the  desired  effect  that  this  line  of  color  should 
not  dry,  even  partially,  until  the  tint  on  the  whole  side  of  the  lever  has  been 
put  on,  it  will  be  as  well  to  draw  the  pen  again  very  lightly  over  the  same 
part,  so  that  the  line  may  retain  as  much  tint  as  possible.  Immediately  this 
has  been  done,  the  brush,  properly  filled  with  the  same  tint,  is  to  pass  along 
and  join  the  inner  edge  of  this  narrow  strip  of  color,  and  the  whole  surface  of 
the  lever  filled  in.  Thus  a  distinct  and  regular  line  of  light  is  obtained,  and, 
in  fact,  the  lever,  or  whatever  else  the  object  may  be,  covered  in  a  shorter  time 
than  usual.  A  still  more  expeditious  way  of  coloring  such  surfaces  is  to  draw 
a  second  line  of  color  along  and  joining  the  opposite  edge  of  the  lever  or  other 
object,  and  then  expeditiously  to  fill  in  the  intermediate  space  between  the  two 
wet  lines  by  means  of  the  brush.  By  similar  means  the  line  of  light  on  a 
cylinder,  shaft,  or  other  circular  body,  may  be  beautifully  expressed.  To  indi- 
cate this  light  with  perfect  regularity  is  highly  important,  for,  if  a  strict  uni- 
formity be  not  maintained  throughout  its  whole  length,  the  object  will  look 
crooked  or  distorted.  After  having  marked  in  pencil,  or  guessed  the  position 
of  the  extreme  light,  take  the  drawing-pen,  well  filled  with  a  just  perceptible 
tint,  and  draw  a  line  of  color  on  one  side  the  line  of  light,  and  almost  touch- 
ing it ;  then  with  the  brush,  filled  with  similar  light  tint,  join  this  line  of  color 
while  still  wet,  and  fill  up  the  space  unoccupied  by  the  shade-tint,  within 
which  the  very  light  color  in  the  brush  will  disappear.  Let  that  part  of  the 
object  on  the  other  side  of  the  line  of  light  be  treated  in  the  same  way,  and 
the  desired  effect  of  a  stream  of  light  clear  and  mathematically  regular  will  be 
obtained.  The  extreme  depth  of  shade,  as  well  as  the  line  of  light  in  such  rods, 
may,  with  great  effect,  be  indicated  by  filling  the  pen  with  dark  shade-tint,  and 
drawing  it  exactly  over  the  line  representing  the  deepest  part  of  the  shade. 
On  either  side  and  joining  this  strip  of  dark  color,  another,  composed  of 
lighter  tint,  is  to  be  drawn.  Others  successively  lighter  are  to  follow,  until, 
on  one  side,  the  line  of  the  rod  is  joined,  and  on  the  other  the  lightest  part  of 
the  rod  is  nearly  reached.  The  line  of  light  is  then  to  be  shown,  and  the  faint 
tint  used  on  this  occasion  spread  with  the  brush  lightly  over  the  whole  of  that 
part  of  the  rod  situated  on  either  side  of  this  line,  thus  blending  into  smooth 
rotundity  the  graduated  strips  of  tint  drawn  by  the  pen. 


136  SHADES  AND  SHADOWS. 

In  all  tinted  drawings  the  more  important  parts,  whether  the  machinery  or 
the  structure,  should  be  more  conspicuously  expressed  than  those  parts  which 
are  mere  adjuncts.  Thus,  if  the  drawing  be  to  explain  the  construction  of 
the  machine,  the  tint  of  edifice  and  foundations  may  be  kept  lighter  and  more 
subdued  than  those  of  the  machine  ;  and  if  the  machine,  on  the  contrary,  be 
unimportant,  it  may  be  represented  quite  light,  or  in  mere  outline,  while  the 
edifice  is  brought  out  conspicuously. 

With  regard  to  washings,  the  soft  sponge  is  an  implement  not  to  be  neg- 
lected by  the  draughtsman  ;  it  is  an  excellent  means  of  correcting  great  errors 
in  drawing,  better  than  rubber  or  an  eraser,  but  care  of  course  must  be  taken 
to  wash  and  not  to  rub  off  the  surface,  and  for  errors  in  coloring  washing  is 
almost  the  only  corrector.  In  removing  or  softening  color  on  large  surfaces, 
the  sponge  is  to  be  used,  and  for  small  spots  the  brush.  While  coloring,  keep 
a  clean,  moist  brush  by  you  :  it  will  be  extremely  useful  in  removing  or  modi- 
fying a  color. 

The  immediate  effect  of  washing  is  to  soften  a  drawing,  an  effect  often  very 
desirable  in  architectural  and  mechanical  drawings,  and  the  process  is  simple 
and  easily  acquired ;  keep  the  sponge  or  brush  and  water  used  clean  ;  after  the 
washing  is  complete,  take  up  the  excess  of  moisture  by  the  sponge  or  brush,  or 
by  a  piece  of  clean  blotting-paper.  Where  great  vigor  is  required,  let  the 
borders  of  the  different  tints  be  distinct. 

There  are  no  conventional  tints  that  draughtsmen  have  agreed  upon  to  be 
uniformly  used,  to  represent  different  materials.  India  ink  is  not  a  black,  but 
a  brown,  making  with  a  blue  a  greenish  cast,  and  with  gamboge  a  smear.  A 
colored  drawing  is  better  without  the  use  of  India  ink  at  all ;  any  depth  of 
color  may  be  as  well  obtained  with  blue  as  with  black  ;  there  is  also  an  objec- 
tion to  gamboge,  that  it  is  gummy,  and  does  not  wash  well,  and  the  effect  is 
better  obtained  with  yellow  ochre.  For  the  reds,  the  madder  colors  are  the 
best,  as  they  stand  washing  ;  for  the  shade-tint  of  almost  every  substance  a 
neutral  tint,  Payne's  gray,  or  madder  brown  subdued  with  indigo. 


PLOTTING. 

PLOTTING  is  the  laying  out  on  paper  in  plan  or  in  horizontal  projection 
the  boundaries  of  lots,  estates,  farms,  etc.,  portions  of  the  earth's  surface  of 
greater  or  less  extent,  from  the  notes  of  surveys  or  other  records.  When  the 
extents  are  large,  beyond  the  usual  limits  of  personal  property,  and  embracing 
degrees  of  latitude  and  longitude,  the  plots  are  designated  as  maps  ;  but  if  of 
small  extent — as  lots,  estates,  and  farms — they  are  usually  designated  as  plans 
or  plots.  After  completing  the  outlines,  it  is  usual  to  fill  up  the  plot,  with  the 
characteristic  features,  geographical,  geological,  agricultural,  industrial,  and 
domestic,  which  are  expressed  more  or  less  conventionally,  as  will  be  shown 
under  the  head  of  "  Topographical  Drawing." 

Scales. — The  choice  of  the  scale  for  the  plot  depends  in  a  great  measure  on 
the  purpose  for  which  the  plan  is  intended.  It  should  be  large  enough  to 
express  all  the  details  desirable,  modified  by  the  circumstances  whether  the 
map  is  to  be  portable  or  whether  space  can  be  afforded  for  the  exhibition  of 
a  large  plan.  We  must  adapt  our  plan  for  the  purposes  which  it  is  intended 
to  illustrate,  and  the  place  it  is  to  occupy. 

Plans  of  house-lots  are  usually  named  as  being  so  many  feet  to  the  inch  ; 
plots  of  farm-surveys,  as  so  many  chains  to  the  inch  ;  maps  of  surveys  of 
States,  as  so  many  miles  to  the  inch  ;  and  maps  of  railway-surveys,  as  so  many 
feet  to  the  inch,  or  so  many  inches  to  the  mile. 

Formerly  the  lines  of  farms  were  measured  by  the  four-rod  chain  ;  latterly 
the  100-foot  chain  is  more  usually  adopted.  Two  to  three  chains  to  the  inch 
was  then  a  very  common  scale. 

State  surveys  are  of  course  plotted  on  a  smaller  scale  than  those  of  farms. 
On  the  United  States  Coast  Survey  all  the  scales  are  expressed  fractionally  and 
decimally.  The  original  surveys  are  generally  on  a  scale  of  one  to  ten  or  twenty 
thousand,  but  in  some  instances  the  scale  is  larger  or  smaller.  The  public  sur- 
veys embrace  three  general  classes  :  1.  Small  harbor-charts.  2.  Charts  of  bays, 
sounds,  etc.  3.  General  coast-charts. 

The  scales  of  the  first  class  vary  from  1 :  5,000  to  1 :  60,000,  according  to  the 
nature  of  the  harbor  and  the  different  objects  to  be  represented. 

The  scale  of  the  second  class  is  usually  fixed  at  1 : 80,000.  Preliminary 
charts,  are,  however,  issued  of  various  scales,  from  1 :  80,000  to  1 : 200,000. 

Of  the  third  class  the  scale  is  fixed  at  1 :  400.000  for  the  general  chart  of  the 
coast  from  Gay  Head  to  Cape  Henlopen,  although  considerations  of  the  prox- 
imity and  importance  of  points  on  the  coast  may  change  the  scales  of  charts  of 
other  portions  of  our  extended  coast. 


138  PLOTTING. 

On  all  plots  of  large  surveys,  it  is  very  desirable  that  the  scales  adopted 
should  bear  a  definite  numerical  proportion  to  the  linear  measurement  of  the 
ground  to  be  mapped,  and  that  this  proportion  should  be  expressed  fractionally 
on  the  plan,  even  if  the  scale  be  drawn  or  expressed  some  other  way,  as  chains 
to  the  inch.  The  decimal  system  has  the  most  to  recommend  it,  and  is  gener- 
ally adopted  in  government  surveys. 

For  railroad-surveys,  the  New  York  general  railroad  law  directs  the  scale  of 
map  which  is  to  be  filed  in  the  State  Engineer's  office,  to  be  500  feet  to  one 
tenth  of  a  foot,  1  :  5,000. 

For  the  canal-maps,  a  scale  of  two  chains  to.  the  inch,  1 : 1,584  is  employed. 
In  England,  plans  and  sections  for  projected  lines  of  inland  communication,  or 
generally  for  public  works  requiring  the  sanction  of  the  Legislature,  are  re- 
quired, by  the  "standing  orders,"  to  be  drawn  to  scales  not  less  than  four 
inches  to  the  mile,  1 : 15,840,  for  the  plan,  and  100  feet  to  the  inch,  1 : 1,200, 
for  the  profiles. 

In  the  United  States  engineer  service  the  following  scales  are  prescribed  : 

General  plans  of  buildings 10  feet  to  the  inch,     :  120 

Maps  of  ground  with  horizontal  curves  1  foot  apart    .         .  50      "                 "      1 :  600 

Topographical  maps  li  mile  square 1  mile  to  2  feet,      :  2,640 

Topographical  maps  comprising  3  miles  square  .         .         .  1        "        1  foot,      :  5,280 

Topographical  maps  comprising  between  4  and  8  miles        .  1        "        6  in.,        :  10,560 

Topographical  maps  comprising  9  miles  square  .         .         .  1        "        4      "         :  15,840 

Maps  not  exceeding  24  miles  square 1        "        2      "      1  : 31,680 

Maps  comprising  50  miles  square 1        "        I  inch,  1 :  63,360 

Maps  comprising  100  miles  square 1        "        \      "      1 : 126,720 

Surveys  of  roads  and  canals 50  feet  to  1      "      1 :  600 

In  cities  and  towns,  lots  and  squares  are  generally  rectangular,  and  they  can 
be  readily  plotted  on  any  convenient  scale. 

Fig.  241  is  a  plan  of  the  usual  New  York  city  lot,  25  x  100,  on  a  scale  of 
20  feet  to  the  inch,  or  -%fa  full  size. 

Fig.  242  is  a  city  square  containing  thirty-two  of  these  lots,  on  a  scale  of 
100  feet  to  the  inch,  or  T-gVo-  ^ne  most  accurate  way  is  to  plot  the  large  rec- 
tangle 400  x  200  feet,  and  then  subdivide  it. 

Fig.  243  is  a  plan  of  the  same  city  squares,  with  the  inclosing  streets,  on  a 
scale  200  feet  to  the  inch,  or  •s^. 

But  there  are  many  lots,  and  most  estates,  which  are  not  rectangular,  the 
angles  of  which  are  recorded,  which  must  be  plotted  by  the  aid  of  a  pro- 
tractor. 

If  the  survey  has  been  made  by  triangles,  the  principal  triangles  are  first 
laid  down  in  pencil  by  the  intersection  of  their  sides,  the  length  being  taken 
from  the  scale  and  described  with  compasses.  In  general,  when  the  surveys 
have  been  conducted  without  instruments  to  measure  the  angles,  as  the  com- 
pass or  theodolite,  the  position  of  the  points  on  paper  are  determined  by 
the  intersection  and  construction  of  the  same  lines  as  has  been  done  in 
the  field. 

Surveys  are  mostly  conducted  by  measuring  the  inclination  of  lines  to  a 
meridian  or  to  each  other  by  the  compass  or  by  the  theodolite.  In  the  sur- 


PLOTTING. 


139 


veys  of  farms,  where  great  accuracy  is  not  required,  the  compass  is  most  used. 

The  compass  gives  the  direction  of  a  line  in  reference  to  the  magnetic  meridian. 

The  variation  from  the  true  meridian,  or  a  direct 

north-and-south  line,  varies  considerably  in  different 
parts  of  the  country.  In  1875  the  line  of  variation 
in  which  the  needle  pointed  directly  north,  passed  in 
a  nearly  straight  direction  from  Wilmington,  North 


FIG.  241. 


FIG.  242. 


Carolina,  to  Cleveland,  Ohio.  At  all  places  east  of  this  line  the  variation  is 
westerly,  that  is,  the  needle  points  west  of  the  line  north.  West  of  this  line 
the  variation  is  easterly. 

Fig.  24A  represents  the  plot  of  a  compass  survey,  with  the  positions  of  the 
protractor  in  laying  off  the  angles.  To  the  left  of  the  figure  are  given  the 
field-notes.  In  this  way  of  plotting,  a  meridian  is  laid  off  at  the  intersection 
of  each  set  of  lines.  Sometimes  the  angles  are  plotted  directly  from  the  deter- 
mination of  the  angle  of  deflection  of  two  courses  meeting  at  any  point,  with- 
out laying  down  more  than  one  meridian  (Fig.  245).  When  the  first  letters 
of  the  bearing  are  alike,  that  is,  both  N.  or  both  S.,  and  the  last  letters  also 
alike,  both  E.  or  both  W.,  the  angle  of  deflection  C  B  B'  will  be  the  difference 
of  the  bearings,  or,  in  this  instance^  20°. 


140 


PLOTTING. 


When  the  first  letters  are  alike  and  the  last  different  (Fig.  246),  the  angle 
C  B  B'  will  be  the  sum  of  the  two  bearings. 


j  i 


L 


1   [ 


r 


FIG.  243. 


When  the  first  letters  are  different  and  the  last  alike  (Fig.  247),  subtract 
the  sum  of  the  bearings  from  180°  for  the  angle  C  B  B' ;  when  both  the  first 
letters  and  last  are  different,  subtract  their  difference  from  180°  for  the  angle. 


-(5)- 
3.55 


GO 

-(4>- 
222 

£ 
od 

-(3)- 
1.29 
H 


-(2)- 
2.70 


-dh- 


FIG.  244. 


Instead  of   drawing  a  meridian  through   each   station,  or  laying  off  the 
angle  of  deflection,  by  far  the  easiest  way  is  to  lay  off  but  a  single  meridian 


PLOTTING. 


141 


near  the  middle  of  the  sheet ;  lay  off  all  the  bearings  of  the  survey  from  some 
one  point  of  it,  as  shown  in  Fig.  248,  and  number  to  correspond  with  the  sta- 
tions from  which  the  bearings  are  taken,  and  then  transfer  them  to  the  places 


FIG.  246. 


FIG.  247. 


where  they  are  wanted  by  any  of  the  instruments  used  for  drawing  parallel 
lines.  For  the  protracting  of  the  rough  plan,  sheets  of  drawing-paper  can  be 
bought  with  protractors  printed  on  them.  When  the  plans  are  large,  it  is 


FIG.  249. 


often  convenient  to  lay  out  two  or  three  meridians  on  different  parts  of  the 
sheet,  and  lay  off  the  bearings  of  lines  adjacent  to  each  meridian  upon  them. 

In  plotting  from  a  survey  by  a  theodolite  or  transit,  it  is  generally  usual  to 
lay  off  the  angles  of  deflection  of  the  different  lines  as  taken  in  the  field,  plot- 
ting all  the  tie-lines  as  corrections. 

When  the  plot  of  a  survey  does  not  close — that  is,  come  together,  or  return 
to  the  point  of  commencement,  as  it  seldom  does  exactly — it  may  be  corrected 
or  forced  ;  but  first  be  sure  that  the  bearings  and  distances  as  recorded  are  laid 
down  accurately,  and  then  proceed  to  correct  as  follows  : 

If  the  plot  of  the  last  line  does  not  close  up  the  outline  of  the  figure  exactly, 
by  its  extremity  falling  upon  the  point  of  beginning  of  the  plot,  as  upon  the  point 
a  (Fig.  249),  instead  of  upon  1,  either  the  survey  or  the  plotting  is  incorrect. 


142 


PLOTTING. 


If  the  latter  be  correct,  the  error  of  the  survey  must  be  balanced,  or  distributed 
through  the  lines  and  angles  of  the  plot.  Connect  1  with  a,  and  draw  lines 
parallel  to  1  a  through  2,  3,  4,  5,  of  the  plot.  Draw  an  indefinite  line,  1  1}  (Fig. 
250),  and  on  this,  with  any  convenient  scale,  lay  off  consecutively  the  lines  of  the 
survey,  1-2,  2-3,  3-4,  4-5,  5-a.  Erect  perpendiculars  at  the  extremities  of  the 
lines,  2,  3,  4,  5,  and  b.  On  the  perpendicular  a  b,  lay  off  1  a  from  the  plot  and  con- 
nect 1 1.  The  intersections  of  the  perpendiculars  by  this  line  will  determine  how 


CJ 


much  each  of  the  points  of  the  plot  are  to  be  moved  on  the  parallels  to  1  a  to  dis- 
tribute the  error.     The  dotted  lines  on  the  figure  show  the  corrected  outline. 

By  the  aid  of  the  Traverse  Table  a  survey  may 
be  balanced  and  accurately  plotted.  The  Traverse 
Table  (see  appendix)  is  a  table  of  differences  of 
latitudes  and  departures,  the  difference  of  latitude 
between  two  stations  being  the  difference  north 
and  south  between  them  ;  the  difference  of  depart- 
ure, the  difference  east  and  west. 
s  Thus,  ]\r  S  (Fig.  251)  being  the  meridian,  A  0 
is  the  difference  of  latitude  between  A  and  B,  and 
A  D  the  departure. 

The  differences  vary  according  to  the  length 
of  A  B,  and  the  angle  it  makes  with  the  meri- 
dian. 
Taking  the  field-notes  of  the  previous  survev,  we  make  a  table  as  follows  : 


W- 


STATION. 

Bearing. 

Distance. 

Latitude. 

N.                      S. 

Departure. 
E.                      W. 

1  

N.  35°    E. 

N.  83|°  E. 
S.    57°    E. 
S.    34i°W. 
N.  56|°  W. 

2-70 
1-29 
2'22 
3-55 
3-23 

2  -21 
•15 

1-78 

1-21 
2-93 

1-55 

1-28 
1-86 

2-00 
2-69 

2 

3  

4  

5.  .      .    . 

4-14 

4-14 

4-69 

4-69 

In  the  Traverse  Table,  on  the  line  with  35°,  and  in 
column  2,  latitude  =  1  '638     departure  =  1  '147 
"       7,         "       =    -573  "        =    '401 

2-211  1-548 

Again,  on  the  line  with  83-j-0,  in 

column  1,  latitude  =    -113     departure  =    '994 
=     -0226  "        =     -1987 

"        =     -01019         "        =     -08942 


•14579  1-28212 

And  in  the  same  manner  the  table  is  completed. 


PLOTTING. 


143 


The  table  following  is  constructed  by  adding  up  the  northings  and  sub- 
tracting the  southings  for  the  latitude,  and  by  adding  up  the  eastings  and 
subtracting  the  westings  for  the  departures. 


STATION. 

Total  latitude  from 
station. 

Total  departure  from 
station  1  -. 

1  

o-oo 

o-oo 

2  

+  2-21  N. 

+  1'55  E 

3  

+  2'36  N. 

+  2  '83  E. 

4  

+  ri5  N. 

+  4'69  E. 

6  

—  1'78  S 

+  2  '69  E 

1  

O'OO 

o-oo 

From  this  table  the  survey  can  be  readily  plotted  (Fig.  252).  Draw  the 
meridian  through  the  point  taken  for  station  1  ;  measure  to  the  north  2*21 
chains  to  A  ;  draw  an  easterly  line,  or  one  perpendicular  to  the  meridian  at 
A,  and  lay  off  on  it  1  -55  chains,  and  we  have  station  2  ;  measure  again  from  1 


northerly  2*36  chains  to  B,  and  lay  off  from  B  due  easterly  2 -83  chains  for 
station  3  ;  measure  again  from  1  northerly  1  -15  chains  to  C,  and  lay  off  from 
C  due  easterly  4*69  chains  for  station  4  ;  measure  again  from  1  southerly 
1'78  chains  to  D,  and  layoff  from  D  easterly  2*69  chains  for  station  5. 
Connect  1,  2,  3,  4,  5,  and  1  for  the  complete  plot. 

In  this  survey  the  latitudes  balance,  4'14  to  4 '14,  and  the  departures  bal- 
ance, 4*69  to  4 -69,  but  this  seldom  happens.  Generally  there  is  a  difference 
which  must  be  balanced  before  plotting.  For  instance,  in  this  survey,  had  the 

northings  been  J          and  had  the  southings  been  |  2 -95  the  difference  would 
1*70 

14-05 


144 


PLOTTING. 


have  been  *14  to  be  divided,  in  proportion  to  their  lengths,  between  the  north- 
ings and  the  southings,  adding  to  the  former  and  deducting  from  the  latter. 
The  total  northings  and  southings  is  4'05  -1-  4'19  =  8*24  chains,  in  which  an 
error  of  14  links  is  to  be  balanced,  or  about  -017  chain  to  each  chain.  In 
the  2-20  N.  the  correction  will  be  2  '20  x  -017  =  '0374,  or  about  -04,  and  with- 
out much  calculation  we  can  see  that 

f2'24  p.  22 

the  corrected  northings  will  be  J          and  the  corrected  southings  will  be  -j  %'9Q 


-j     ' 
14*12 


I  4-12 

The  same  calculation  is  applied  to  the  departures  when  there  is  a  difference 
in  the  total  eastings  and  westings. 

The  errors  are  to  be  balanced  before  the  survey  is  plotted. 

When  a  field  has  been  plotted,  it  can  be  divided  into  triangles,  and  its  area  can  be  calculated  ; 
but,  having  the  latitudes  and  departures  balanced  and  tabulated,  the  area  can  be  calculated  as  fol- 
lows: 


STATION. 

Latitude. 

Departure. 

Double 

longitude. 

Double  area. 

N   + 

S.  - 

E.  + 

W.  - 

N.  +        |                 S.  - 

1 

2-21 
•15 

1-78 

1-11 

2-93 

1-55 
1-28 
1-86 

2-00 
2-69 

+  1-55 
+  4'38 
+  7-52 
+  7-38 
+  2-69 

3-4255 
0-6570 

4-7882 

9-0992 
21-6234 

2  

3  

4 

5  

4-14 

4-14 

4-69 

4-69 

8-8707 

30-7226 

8-8707 

Content  =  1A.,  OR.,  15P. 


2)21-8519 


10-9259  = 
1  •  09259  acres. 


The  first  five  columns  are  from  the  preceding  tables.  To  construct  the  column  of  double  longi- 
tudes :  the  double  longitude  of  the  first  course  is  equal  to  its  departure. 

The  double  longitude  of  the  second  course  is  equal  to  the  double  longitude  of  the  first  course,, 
added  to  the  departure  of  that  course,  added  to  the  departure  of  the  second  course. 

The  double  longitude  of  the  third  course  is  equal  to  the  double  longitude  of  the  second  course^ 
added  to  the  departure  of  that  course,  added  to  the  departure  of  the  third  course. 

The  double  longitude  of  any  course  is  equal  to  the  double  longitude  of  the  preceding  course  added 
to  the  departure  of  that  course,  added  to  the  departure  of  the  course  itself ;  the  double  longitude  of 
the  last  course  is  equal  to  its  departure. 

Thus,  the  double  longitude  of  first  course  is  its  departure =  1  '55 

add  the  departure  of  first  course 1'55 

add  the  departure  of  second  course 1  '28 

and  we  have  the  double  longitude  of  second  course =  4-38 

add  the  departure  of  second  course 1'28 

add  the  departure  of  third  course 1'86 

and  we  have  the  double  longitude  of  third  course =  7.52 

add  the  departure  of  third  course 1*86 

subtract  the  departure  of  fourth  course 2.00 

and  we  have  the  double  longitude  of  fourth  course =  7'38 

subtract  the  departure  of  fourth  course 2'00 

subtract  the  departure  of  fifth  course 2'69          4'69 

and  we  have  the  double  longitude  of  fifth  course =  2-69 


PLOTTING. 


145 


Multiply  the  double  longitude  of  each  course  by  the  latitude  of  that  course,  placing  the  north 
products  in  one  column  and  the  south  products  in  another ;  subtract  the  lesser  total  of  the  one 
column  from  the  greater  total  of  the  other, 
and  divide  the  difference  by  two.    The  prod- 
uct will  be  in  square  chains,  which,  divided 
by  ten,  will  give  the  result  in  acres  and 
decimals. 

The  area  of  an  irregular  figure  can  be 
calculated  most  conveniently,  and  with  suf- 
ficient accuracy,  by  dividing  it  into  triangles, 
measuring  the  height  and  base  of  each,  cal- 
culating the.  area  of  each,  and  adding  the 


r 

FIG. 


253. 


areas  together. 

Or,  the  polygon  may  be  resolved  readily 

into  a  single  triangle,  and  its  area  calculated.  For  instance,  take  the  five-sided  polygon,  1,  2,  3,  4,  5 
(Fig.  253).  Call  the  side  5  1  the  base,  and  extend  it.  Join  1  and  3.  Draw  2  1'  parallel  to  1  3. 
Join  1'  and  4.  Draw  2'  3  parallel  to  1'  4.  Join  2'  and  4.  The  triangle  2'  4  5  will  be  a  triangle 
equal  to  the  polygon. 

The  same  construction  will  apply  to  a  figure  of  a  greater  number  of  sides. 

The  area  of  a  triangle  can  be  calculated  graphically  (Fig.  254).     Let  the  scale  be  two  chains  to 
the  inch.     Prepare  a  strip  of  drawing-paper  one  inch  wide,  and  divide  it  by  perpendicular  lines  in 


FIG.  255. 


20ths  of  an  inch.  Apply  it  to  the  triangle  A  B  C  so  that  one  edge  will  fall  upon  A,  and  the  other 
at  B.  Keeping  the  same  points  on  the  extended  line  A'  B,  slide  the  scale  up  till  its  upper  edge 
arrives  at  the  point  C.  The  line  A'  C  in  divisions  of 
the  scale  is  the  area  of  the  triangle  in  square  chains. 

If  the  scale  had  been  three  chains  to  the  inch,  the 
strip  should  have  been  f  of  an  inch  in  width ;  if  four 
chains  to  the  inch,  then  f  of  an  inch  in  width,  and  so  on. 

When  the  lines  of  a  plot  are  irregular,  as  in  Fig. 
255,  draw  across  it  a  number  of  equidistant  parallel 
lines,  and  with  a  strip  of  paper  measure  these  lines,  one  after  another,  till  the  sum  of  their  lengths 
is  marked  on  the  edge  of  the  strip.  Cut  the  strip  at  the  last  mark,  and  fold  it  in  two.  This  measure 
(half  the  length  of  the  strip),  multiplied  by  the  uniform  width  between  the  parallel  lines,  will  give 
very  nearly  the  area. 

Having  completed  the  plot — that  is,  the   main   lines  of   the  survey — the 
filling  of  other  points  may  in  general  be  done  on  paper,  the  same  way  that 
they  have  been  established  in  the  field.     Intersections  of  the  main  lines  by 
10 


146 


PLOTTING. 


roads,  streams,  fences,  and  the  like,  are  measured  off ;  other  points  not  inter- 
secting, are  usually  fixed  by  triangles  or  by  offsets  from  the  main  lines,  or  lines 
run  on  purpose  by  angles  from  the  main  lines. 


fixed  Sea 


FIG.  256. 

In  case  of  unimportant  lines,  as  the  crooked  brook,  for  instance  (Fig.  256), 
offsets  are  taken  to  the  most  prominent  angles,  as,  a,  a,  a,  and  the  intermediate 
bends  are  sketched  by  eye  into  the  field-book.  In  copying  them  on  the  plan  a 
similar  construction  is  adopted. 

The  most  rapid  way  of  plotting  the  offsets  is  by 
the  use  of  a  plotting  and  offset  scale  (Fig.  257),  the 
one  being  fixed  parallel  to  the  line  A  B  from  which 
the  offsets  are  to  be  laid  off,  at  such  a  distance  from 
it,  that  the  zero-line  on  the  movable  scale  coincides 
with  it,  while  the  zero  of  its  own  scale  is  on  a  line 
perpendicular  to  the  position  of  the  station  A  from 
which  the  distances  were  measured.  It  is  to  be  ob- 
served that  in  the  field-book  all  the  measures  are  re- 
ferred to  the  point  of  beginning  on  any  one  straight 
line.  Having  placed  the  plotting-scale,  move  the 
offset-scale  to  the  first  distance  by  the  scale  at  which 
an  offset  has  been  taken,  mark  off  now  on  the  offset- 
scale  the  length  of  the  offset  on  its  corresponding 
side  of  the  line.  Proceed  then  to  the  next  distance, 
establishing  thus  repeated  points,  join  the  points  by 
lines  as  they  are  on  the  ground. 

The  plotting  and  offset  scale  must  of  course  be  of 
the  same  scale  as  the  rest  of  the  drawing,  on  which 
account  it  may  not  always  be  possible  to  obtain  such 
scales  adapted  to  those  of  the  plan  ;  but  they  may  be 
easily  constructed  of  thick  drawing-paper  or  paste- 
board. 

When  a  great  deal  of  plotting  to  one  scale  is 
necessary,  as  in  government  surveys,  the  offset-scale 
may  be  made  to  slide  in  a  groove  upon  the  plotting- 
scale. 

In  protracting  the  triangles  of  an  extended  trigo- 
nometrical survey  in  which  the  sides  have  been  cal- 
culated or  measured,  it  is  better  to  lay  down  the 
triangles  from  the  length  of  their  sides  rather  than 
by  measuring  the  angles,  because  measures  of  length 
can  be  taken  with  more  accuracy  from  a  scale,  and  transferred  to  the  plan 
with  more  exactness  than  angles  can  be  pricked  off  from  a  protractor ;  but, 


PLOTTING. 


147 


for  ordinary  surveys,  the  triangulation  is  most  frequently  and  expeditiously 
plotted  by  the  means  of  a  protractor. 

The  outlines  of  the  survey  having  been  balanced  and  plotted  in,  and  the 
subsidiary  points,  as  established  by  offsets  and  by  triangles,  the  filling  in  of  the 
interior  detail,  with  the  natural  features  of  the  ground,  from  the  skeleton  or 
suggestions  in  the  field-book  or  other  records,  is  done  according  to  imitative 
and  conventional  signs,  to  be  shown  under  "Topographical  Drawing." 

The  public  lands  of  the  United  States  are  surveyed,  mapped,  and  divided 
into  nearly  square  tracts,  according  to  the  following  system  : 

Ranges. — Standard  lines  must  first  be  determined,  from  which  to  measure. 
Accordingly,  in  each  land-district  some  meridian-line  is  run  due  north  and 
south  ;  this  is  called  the  Principal  Meridian.  From  some  point  of  the  Principal 
Meridian  is  also  run  a  line  due  east  and  west,  called  the  Base-Line. 

Other  lines  are  then  run  in  the  same  direction  as  the  Principal  Meridian,  at 
distances  of  six  miles  (measured  on  the  Base-Line)  on  each  side  of  it.  The 
strip  between  the  Principal  Meridian  and  the  first  line  thus  run  east  of  it  is 
known  as  Range  1  East ;  the  second  strip  is  Range  2  East,  etc.  And  so  on 
the  west  ;  the  successive  strips  running  north  and  south,  six  miles  wide,  are 
called  Range  1  West,  Range  2  West,  etc.  This  division  is  shown  in  Fig.  258. 


i 

r 

Tp.2 
North 

be 

. 

few 

i? 

t? 

i3 

i 

I 

Ki 

k* 

1 

Tp.l 

North 

BASE 

LINE 

Tp.l 

a 

f 

«. 

1 

South 

e 

2 
9 

Tp.2 

South 

P 

fe 

•( 

e^ 

-j 

! 

FIG.  258. 


FIG.  259. 


Townships. — In -like  manner,  lines  are  run  north  and  south  of  the  Base- 
Line  at  intervals  of  six  miles.  These  lines  cut  at  right  angles  those  which 
separate  the  ranges,  and  with  them  form  squares  six  miles  on  each  side,  called 
townships.  Each  township  contains  thirty-six  square  miles. 

The  township  nearest  the  Base-Line  on  the  north  is  known  as  Township  1 
North,  of  whatever  range  it  may  be  in  ;  the  next  farther  north  is  Township  2 
North,  of  that  range — and  so  on.  In  like  manner,  going  south  from  the 
Base-Line,  we  have  in  succession  Township  1  South,  Township  2  South,  etc. 
(Fig.  259). 


148 


PLOTTING. 


Sections. — Each  township  is  divided  into  thirty-six  squares,  called  Sections, 
each  one  mile  long  and  one  mile  wide,  and  therefore  having  an  area  of  one 
square  mile.  The  sections  of  a  township  are  numbered  1,  2,  3,  etc.,  up  to  36, 
beginning  at  the  northeast,  and  running  alternately  from  right  to  left  and  from 
left  to  right,  as  shown  in  Fig.  260. 


6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

1  mile. 


E 


FIG.  260. 


FIG.  261. 


A  section  may  be  subdivided  into  half-sections,  quarter-sections,  eighths, 
and  sixteenths,  designated  as  in  the  example  that  follows  : 

Let  F  G  (Fig.  261)  be  Section  3  of  Township  2  North,  in  Range  1  West ; 
then — 

A  is  N.  (north)  -J  of  Section  3,  Township  2  North,  Range  1  West. 

B  is  S.  W.  (southwest)  ±  of  Section  3,  Township  2  North,  Range  1  West, 

C  is  W.  (west)  |  of  S.  E.  (southeast)  i  of  Section  3,  Township  2  North, 
Range  1  West. 

D  is  N.  E.  i  of  S.  E.  i  of  Section  3,  Township  2  North,  Range  1  West. 

E  is  S.  E.  i  of  S.  E.  i  of  Section  3,  Township  2  North,  Range  1  West. 

Correction- Lines. — If  the  north-and-south  (meridian)  lines  were  parallel  to 
each  other,  the  townships  and  sections  would  be  exact  squares.  But  as  these 
lines  gradually  converge  toward  the  north,  meeting  at  the  pole,  the  townships 
deviate  somewhat  from  squares,  being  narrower  on  the  north  than  on  the  south  ; 
and  the  northern  sections  of  a  township  are  a  little  smaller  than  the  southern 
ones. 

In  order  that  the  townships  of  a  range  may  not  thus  keep  getting  smaller 
and  smaller  as  we  go  toward  the  north,  a  new  base-line,  called  a  Correction- 
Line,  is  taken  at  intervals  (differing  in  length  in  different  land-districts),  and 
new  north-and-south  lines  are  run  at  distances  of  six  miles  measured  on  the 
Correction-Lines. 

The  system  of  survey  described  above  is  not  used  in  Texas,  the  public  lands 
there  being  State  property. 


TOPOGRAPHICAL    DRAWING. 

TOPOGBAPHICAL  DRAWING  is  the  delineation  of  the  surface  of  a  locality, 
with  the  natural  and  artificial  objects,  as  houses,  roads,  rivers,  hills,  etc.,  upon 
it  in  their  relative  dimensions  and  positions,  giving,  as  it  were,  a  miniature 
copy  of  the  farm,  field,  district,  etc.,  as  it  would  be  seen  by  the  eye  moving 
over  it.  Many  of  the  objects  thus  to  be  represented  can  be  defined  by  regular 
and  mathematical  lines,  but  many  other  objects,  from  their  irregularity  of  out- 
line, it  would  be  very  difficult  thus  to  distinguish  ;  nor  are  the  particular 
irregularities  necessary  for  the  expression.  Certain  conventional  signs  have 


FIG.  262. 


FIG.  263. 


FIG.  264. 


FIG.  265. 


therefore  been  adopted  in  general  use  among  draughtsmen,  some  of  which 
resemble,  in  some  degree,  the  objects  for  which  they  stand,  while  others  are 
purely  conventional.  These  signs  may  be  expressed  by  lines,  or  by  tints,  or 
by  both. 

Figs.  262  and  263  represent  meadow  or  grass  land,  the  short  lines  being 
supposed  to  represent  tufts  of  grass  ;   the  bases  of  the  tufts  should  always 


ft 


I  I 

li! 

I! 

Ill  rj 
LU.il 


FIG.  266. 


FIG.  267. 


FIG. 


be  parallel  to  the  base  of  the  drawing,  whatever  may  be  the  shape  of  the  in- 
closure. 

Figs.  264,  265,  266,  267,  give  various  methods  of  representing  trees.  Figs. 
264  and  265  represent  in  plan  a  forest  and  an  orchard,  while  Figs.  266  and  267 
show  the  same  in  elevation.  The  latter  method  of  representing  trees  is  not 


150 


TOPOGRAPHICAL   DRAWING. 


consonant  with  the  projection  of  the  plan,  but  to  many  is  more  expressive  and 
intelligible. 

Fig.  268  represents  cultivated  land.  The  lines  are  supposed  to  represent 
plow-furrows,  and  adjacent  fields  should  be  distinguished  from  each  other  by 
different  inclinations  of  lines. 

Figs.  269  and  270  represent  marsh  or  bog  land.  Fig.  269  is  the  more  ordi- 
nary mode  of  representing  fresh-water  bog,  and  Fig.  270  of  salt-marsh. 


FIG.  269. 


FIG.  270. 


FIG.  271. 


Fig.  271  represents  a  river,  with  mud  and  sand  banks.  Sand  is  designated 
by  fine  dots,  made  with  the  point  of  the  pen  ;  mud  in  a  similar  way,  but  the 
dots  should  be  much  closer  together.  Gravel  is  represented  by  coarser  dots, 
and  stones  by  irregular  angular  forms. 

Water  is  almost  invariably  represented  in  the  same  way,  except  in  connec- 
tion with  bogs,  by  drawing  a  line  parallel  to  the  shore,  following  its  wind- 
ings and  indentations  closely ;  then  another  parallel  a  little  more  distant ;  a 


FIG.  272. 


FIG.  273. 


third  still  more  so  ;  and  so  on.  Brooks,  and  even  rivers,  when  the  scale  is 
small,  are  represented  by  one  or  two  lines.  Fig.  272  gives  a  plan  and  sec- 
tional view  of  water,  in  which  the  white  curves  represent  the  character  and 
direction  of  the  flow  of  streams,  retarded  at  bottom  and  sides,  and  more  rapid 


TOPOGRAPHICAL  DRAWING. 


151 


near  the  surface  and  at  center,  therefore  convex  down  stream.  The  direction 
of  the  current  may  also  be  shown  by  arrows,  as  in  Fig.  271. 

Fig.  273  represents  a  bold  shore  bounded  by  cliffs. 

Fig.  274  represents  a  turnpike.  If  the  toll-bar  and  marks  for  a  gate  be 
omitted,  it  is  a  common  highway.  Fig.  275  represents  a  road  as  sunk  or  cut 


3M  Bar 


FIG.  274. 


FIG.  275. 


FIG.  276. 


FIG.  277. 


through  a  hill.  Fig.  276,  one  raised  upon  an  embankment.  Fig.  277  is  a 
railroad,  often  represented  without  the  cross-ties  by  two  heavy  parallel  lines, 
sometimes  by  but  one. 


FIG.  278. 


FIG.  279. 


FIG.  280. 


FIG.  281. 


A 

Saw-mill, 

«•  - 

Wind-mill,  ( 

& 

Steam-mill, 

m 

Furnace, 

ml 

Woolen-factory, 

& 

Cotton-factory, 

t 

Dwellings,       f§ 

X 

Churches,        m 

^% 

O 

Grave-yards, 

Fig.  278  represents  a  bridge  with  a  single  pier.     Fig.  279,  a  swing  or  draw 
bridge.     Fig.  280,  a  suspension  bridge,  and  Fig.  281  a  ford.     Fig.  282,  a  lock 
of  a  canal.     Canals  are  represented  like  roads,  except 
that  in  the  latter  the  side  from  the  light  is  the  shaded 
line  ;  in  the  former,  the  side  to  the  light.  FlG-  282- 

The  more  important  objects  that  are  likely  to  need  representation  on  a  map 
have  conventional  signs,  as  follows  : 

Signal  of  Survey, 

Telegraph, 

Court-house, 

Post-office, 

Tavern, 

Blacksmith's  shop, 

Guide-board, 

Quarry, 

Grist-mill, 

The  localities  of  mines  may  be  represented  by  the  signs  of  the  planets, 
which  were  anciently  associated  with  the  various  metals,  and  a  black  circle  for 


152 


TOPOGRAPHICAL  DRAWING. 


coal.     Thus,    $   Mercury,    ?    Copper,    ^    Lead,    D  Silver,  O  Gold,    6  Iron, 
K  Tin,  •  Coal. 

The  Representation  of  Hills. — The  two  methods  in  general  use  for  rep- 
resenting with  a  pen  or  pencil  the  slopes  of  ground,  are  known  as  the  vertical 
and  horizontal.  In  the  first  (Fig.  283),  the  strokes  of  the  pen  follow  the  course 
that  water  would  take  in  running  down  these  slopes.  In  the  second  (Fig.  284), 


FIG.  283. 


FIG.  284. 


they  represent  horizontal  lines  traced  round  them,  such  as  would  be  shown  on 
the  ground  by  water  rising  progressively  by  stages,  1,  2,  3,  4,  5,  6,  up  the  hill. 
The  last  is  the  more  correct  representation  of  the  general  character  and  features 
of  the  ground,  and,  when  vertical  levels  or  contours  have  been  traced  by  level 
at  equal  vertical  distances  over  the  surface  of  the  ground,  they  should  be  so 
represented  ;  or  when,  by  any  lines  of  levels,  these  contours  can  be  traced  on 
the  plans  with  accuracy,  the  horizontal  system  should  be  adopted  :  but  where, 
as  in  most  plans,  the  hills  are  but  sketched  in  by  the  eye,  the  vertical  system 
should  be  adopted  ;  it  affords  but  proximate  data  to  judge  of  the  slope,  whereas, 
by  the  contour  system,  the  slope  may  be  measured  exactly.  It  is  a  good  maxim 
in  topographical  drawing  not  to  represent  as  accurate  anything  which  has  not 
been  rigorously  established  by  surveys.  On  this  account,  for  general  plans, 
when  the  surface  of  the  ground  has  not  been  leveled,  nor  is  required  to  be 
determined  with  mathematical  precision,  we  prefer  the  vertical  to  the  hori- 
zontal system  of  representing  slopes. 

On  drawing  hills  on  the  vertical  system,  it  is  very  common  to  draw  contour- 
lines  in  pencil  as  guides  for  the  vertical  strokes.  If  the  horizontal  lines  be 
traced  at  fixed  vertical  intervals,  and  vertical  strokes  be  drawn  between  them 
in  the  line  of  quickest  descent,  they  supply  a  sufficiently  accurate  representa- 
tion of  the  face  of  the  country  for  ordinary  purposes.  It  is  usual  to  make  the 
vertical  strokes  heavier  the  steeper  the  inclination,  and  systems  have  been  pro- 
posed and  used,  by  which  the  inclination  is  defined  by  the  comparative  thick- 
ness of  the  line  and  the  intervening  spaces. 


TOPOGRAPHICAL  DRAWING. 


153 


In  describing  ground  with  the  pen,  the  light  is  generally  supposed  to  de- 
scend in  vertical  rays,  and  the  illumination  received  by  each  slope  is  dimin- 
ished in  proportion  to  its  divergence 
from  the  plane  of  the  horizon.  Thus, 
in  Fig.  285,  it  will  be  seen  that  a  hori- 
zontal surface  receives  an  equal  por- 
tion of  light  with  the  inclined  surface 
resting  upon  it,  and,  as  the  inclined 


FIG.  285. 


surface  is  of  greater  extent,  it  will  be 
darker  than  the  horizontal  in  propor- 
tion to  the  inclination  and  consequent  increase  of  the  surface,  and  on  this 
principle  varied  forms  of  ground  are  represented  by  proportioning  the  thick- 
ness of  stroke  to  the  steepness  of  the  slope. 


FIG.  286. 

In  the  German  system,  as. proposed  by  Major  Lehmann,  of  representing  the 
slopes  of  ground  by  a  scale  of  shade,  the  slope  at  an  angle  of  45°,  as  reflecting 
its  light  horizontally,  is  supposed  to  be  the  greatest 
ever  required  to  be  shown,  and  is  represented  by  black, 
while  the  horizontal  plane  reflecting  all  rays  upward  is 
represented  by  white.  Fig.  286  gives  the  intervening 
proportions  of  black  and  white. 

A  modification  of  Lehmann's  method,  proposed  by 
the  United  States  Coast  Survey,  has  the  advantage  of 
discriminating  between  slopes  of  greater  inclination 
than  45°.  The  table  gives  the  proportions  of  black 
and  white  for  different  inclinations,  and  the  construc- 
tion may  easily  be  understood  from  Fig.  287. 

Contour- Lines. — Conceive  a  hill  to  be  completely  covered  with  water. 
Then  suppose  the  water  to  be  drawn  down,  say  five  feet  at  a  time.  Each  line 
of  contact  of  the  hill  and  the  water  will  be  a  contour-line,  or  a  line  every  point 
of  which  is  at  the  same  height  or  level  above  a  fixed  horizontal  plane,  called 
the  datum-plane.  For  a  small  hill,  stake  out  the  ground  in  squares  of  say 
fifty  feet  to  the  side,  and  take  levels  at  each  point  of  these  squares,  and  as  many 
intermediates  as  the  change  of  slope  makes  necessary.  To  draw  the  map,  lay  off 
these  squares  to  a  scale,  and  mark  the  elevation  of  each  point  and  the  interme- 
diates in  pencil.  Then  by  the  eye  draw  in  the  contours  at  such  vertical  dis- 


Slope. 

Proportion  of 
Black.  White. 

24°  or  2£° 

1 

10 

5°  or    6° 

2 

9 

10°  or  11° 

3 

8 

15°  or  16° 

4 

1 

25°  or  26° 

5 

6 

35° 

6 

5 

45° 

7 

4 

60° 

8 

3 

75° 

9 

2 

154 


TOPOGRAPHICAL  DRAWING. 


tances  apart  as  the  requirements  of  the  map  call  forth.    For  a  large  survey,,  say 
of  a  mountain,  such  a  method  is  impracticable.     In  this  case,  the  surveyor 


FIG.  287. 

fixes  a  number  of  points  at  the  same  level,  the  points  being  absolutely  estab- 
lished by  the  transit  or  compass  so  that  they  can  be  plotted  accurately.  Con- 
nect all  points  at  the  same  level,  and  fill  in  the  distances  between  by  the  eye, 
on  the  supposition  that  the  slope  is  uniform  between  these  lines.  The  lines 
absolutely  established  and  those  merely  sketched  in  must  not  be  confounded, 
and  should  be  distinguished  apart  either  by  color,  by  size  of  lines,  or  by  dot- 
ting. The  contour-lines  denoting  every  even  five,  ten,  etc.,  feet  above  the 
datum  or  plane  of  reference  may  be  numbered  with  such  height.  This  is  an 
effective  way  of  representing  hills,  but  is  only  to  be  recommended  when  lines 


FIG.  288. 

have  been  traced  and  it  becomes  a  record  of  facts.  Fig.  288  represents,  on 
double  the  scale,  the  half  of  the  hill,  Fig.  284,  with  one  half  completed  by 
drawing  the  intermediate  contour  lines. 

The  objection  to  the  drawing  of  hills  by  any  system  is  that  the  depths  of 
shade  representing  different  slopes  conflict  with  the  lights  and  shades  of  the 
drawing,  and  are  therefore  confusing.  The  plan  adopted  by  Von  Eggloffstein 
in  his  maps  was  to  form  a  model  and  then  put  in  the  hills  as  they  appeared, 
with  the  rays  of  light  inclined  45°  to  the  plan  of  the  drawing.  He  adopted  a 
ready  way  of  forming  his  model.  The  contours  were  cut  out  of  sheet-wax 
under  the  needle  of  a  sewing-machine,  then  properly  superimposed  on  one 
another.  A  mold  was  then  taken  from  them  in  plaster.  A  model  from  the 
mold,  also  in  plaster,  was  then  taken.  This  was  watered  while  fresh  by  a  verti- 
cal rain  from  a  water-pot,  which  broke  down  the  vertical  edge  of  the  contours, 
and  gave  natural  lines  of  water  shed.  This  model  would  then  be  photographed 


TOPOGKAPHICAL  DRAWING. 


155 


FIG.  289. 


156 


TOPOGRAPHICAL  DRAWING. 


Degree. 

Radii,  ft. 

Central 
Ordinate. 

1° 

5729-65 

0-218 

2° 

2864-93 

0-436 

3° 

1910-08 

0-655 

4° 

1432-69 

0-873 

5° 

1146-28 

1-091 

6° 

955-37 

1-309 

7° 

819-02 

1-528 

8° 

716-78 

1-746 

9° 

637-27 

1-965 

10° 

573-69 

2-183 

under  an  inclined  light,  and  gave  an  admirable  projection.  When  a  model 
was  not  made,  the  hills  are  represented  in  the  same  way  under  an  inclined 
light  of  45°. 

Fig.  289  is  a  map  of  the  harbor  and  city  of  New  Haven,  reduced  from  the 
charts  of  the  United  States  Coast  Survey. 

Plate  VI  is  a  map  of  a  farming  country.     These  two  maps  illustrate  the 
practical  applications  of  topographical  conventionalities. 

Railway  surveys  are  usually  plotted  by  tangents.    The  curves  are  then  put 
in,  and  the  topographical  features  for  the  width  necessary.     The  curves  are 

designated  by  degrees,  as  a  curve  of  1°,  2°,  3°,  etc., 
according  as  the  angle  subtended  at  the  center  by  a 
100-feet  chord  is  1°,  2°,  3°,  etc. 

Knowing  the  tangent  points,  it  is  easy  to  plot  in 
the  curve,  as  the  center  of  the  curve  must  be  the 
intersection  of  the  perpendiculars  to  the  tangents  at 
these  points.  .Or,  if  we  know  one  point  of  tangency 
and  the  radius,  erect  a  perpendicular  at  this  point, 
and  lay  off  the  radius  on  it  to  get  the  center  of  the 
curve. 

When  the  curves  are  larger  than  can  be  described 

by  the  dividers  or  beam  compasses,  they  can  be  plotted  as  shown  in  geometrical 
problems,  or  points  of  a  curve  may  be  obtained  by  calculation  of  their  ordinates, 
and  the  curves  drawn  from  point  to  point  by  sweeps  and  variable  curves.  Ap- 
proximately, knowing  the  central  ordinate  of  the  curve  between  two  points,  the 


-X 


31.43FcdFallpcrMilt 


Level 


FIG.  290. 


central  ordinate  of  one  half  that  curve  will  be  one  quarter  of  the  first ;  but  it 
should  be  observed  that,  the  greater  the  number  of  degrees  in  the  arc,  the  less 
near  to  the  truth  is  the  rule. 

Fig.  291  represents  a  plot  of  a  railway  line  ;  in  this  plot  the  curve  is  repre- 
sented as  a  straight  line,  the  radius  of  curvature  being  written  in.  This  method 
is  sometimes  adopted  when  it  is  desirable  to  confine  the  plot  within  a  limited 


!AL  DRA 


TOPOGRAPHICAL  DRAWING.  157 

' 

space  upon  the  sheet,  and  it  is  convenient  when  plotted  thus  directly  beneath 
the  profile  or  longitudinal  section  (Fig.  290). 

In  plotting  the  section,  a  horizontal  or  base  line  is  drawn  on  which  are  laid 
off  the  stations  or  distances  at  which  levels  have  been  taken  ;  at  these  points  per- 
pendiculars or  ordinates  are  erected,  and  upon  them  are  marked  the  heights  of 
the  ground  above  the  base,  and  the  marks  are  joined  by  straight  lines.  To 
express  rock  in  a  cut,  it  is  generally  represented  by  diagonal  lines  ;  rivers  are 
represented  in  section  by  cross-lines  or  colored  in  blue  ;  a  mud-bottom  by 
masses  of  dots. 

Since  it  would  be  in  general  impossible  to  express  the  variations  of  the  sur- 
face of  the  ground  in  the  same  scale  as  that  adopted  for  the  plan,  it  is  usual 
therefore  to  make  the  vertical  scale  larger  than  that  of  the  horizontal,  usually 
in  proportion  of  10  or  20  to  1.  Thus,  if  the  horizontal  scale  of  the  plan  be  400 
feet  to  the  inch,  the  vertical  scale  would  be  40  or  20  feet  to  the  inch. 

For  the  purpose  of  facilitating  the  plotting  of  profiles,  profile-paper  can  be 
obtained  from  stationers,  on  which  are  printed  horizontal  and  vertical  lines ; 
the  horizontal  lines  being  ruled  at  a  distance  of  -£$  of  an  inch  from  each  other, 
every  fifth  line  being  coarser,  and  every  twenty-fifth  still  heavier  than  the 
others.  Each  of  the  spaces  is  usually  considered  one  foot.  The  vertical  lines 
are  one  quarter  of  an  inch  distant  from  each  other,  every  tenth  line  being 
made  more  prominent  than  the  others  ;  these  spaces  in  general  represent  a 
distance  of  100  feet,  the  usual  distance  between  stations  011  a  railroad.  Much 
time  is  saved  by  the  use  of  this  paper,  both  in  plotting,  and  in  reading  the 
measurements  after  they  are  plotted. 

In  the  plotting  of  sections  across  the  line,  which  are  extended  but  little 
beyond  the  line  of  the  cut  or  embankment,  equal  vertical  and  horizontal  scales 
are  adopted  ;  these  plots  are  mostly  to  determine  the  position  of  the  slope,  or 
to  assist  in  calculating  the  excavation.  To  facilitate  these,  cross-section  paper 
is  sold,  ruled  with  vertical  and  horizontal  lines,  forming  squares  of  TV  of  an 
inch  each.  Every  fifth  line  in  each  direction  is  made  prominent.  When 
cross-sections  are  extended  to  show  the  grade  of  cross-road,  or  changes  of  level 
at  considerable  distance  from  the  line  of  rail,  the  same  scales,  vertical  and  hori- 
zontal, are  adopted  as  in  the  longitudinal  section  or  profile. 

It  will  be  observed,  in  Fig.  290,  that  the  upper  or  heavy  line  represents  the 
line  of  the  rail,  the  grades  being  written  above ;  this  is  the  more  usual  way, 


FIG.  292. 


but  sometimes,  as  in  Fig.  292,  the  profile  and  plan  are  combined  ;  that  is,  the 
heights  and  depths  above  and  below  the  grade-line  of  the  road  are  transferred 
to  the  plan,  and  referred  to  the  line  in  plan,  which  becomes  thus  a  representa- 
tion both  in  plan  and  elevation. 


158 


TOPOGRAPHICAL   DRAWING. 


Cross-sections,  for  grades  of  cross-roads,  etc.,  are  usually  plotted  beneath  or 
above  the  profile  ;  they  may,  if  necessary,  be  plotted  across  the  line  when  plan 
and  profile  are  combined. 

Besides  the  complete  plans  as  above,  giving  the  details  of  the  location,  land 
plans,  so  called,  are  required,  showing  the  position  and  direction  of  all  lines 
of  fences  and  boundaries  of  estates,  with  but  very  few  of  the  topographical  feat- 


TIG.  293. 

ures.  The  center  line  of  the  road  is  represented  in  bold  line,  and  at  each  side, 
often  in  red,  are  represented  the  boundaries  required  for  the  purposes  of  way. 
In  general,  a  width  of  100  feet  is  the  amount  of  land  set  off,  lines  parallel  to 
the  central  line  being  at  a  distance  of  50  feet  on  each  side  ;  but  when,  owing 
to  the  depth  of  the  cut  or  embankment,  the  slopes  run  out  beyond  this  limit, 
the  extent  is  determined  by  plotting  a  cross-section  and  transferring  the  dis- 
tances thus  found  to  the  plan,  and  inclosing  all  such  points  somewhat  within 


TOPOGRAPHICAL  DRAWING. 


159 


the  limits  as  set  off  for  railway  purposes.  These  plans  are  generally  filed  in 
the  register's  office  for  the  county  through  which  the  line  passes. 

Hydrometrical  or  Marine  Surveys. — In  plotting  hydrometrical  or  marine 
surveys,  the  depths  of  soundings  are  seldom  expressed  by  sections,  but  by 
figures  written  on  the  plan,  expressing  the  sounding  or  depth  below  a  datum- 
line,  generally  that  of  high  water.  The  low-water  line  is  usually  represented 
by  a  single  continued  line.  The  soundings  are  generally  expressed  in  fathoms, 
sometimes  in  feet. 

Fig.  293  is  a  map  of  Cape  Cod  Bay  plotted  by  this  method.  The  depths 
are  expressed  in  fathoms  (six  feet),  and  the  dotted  lines  inclose  depths  between 
certain  fixed  limits  so  as  to  plainly  indicate  a  channel  or  bar,  as  the  case  may  be. 

Another  and  an  exceedingly  effective  way  of  making  a  marine  chart  is  to 
express  the  different  depths  by  lines  varying  in  direction,  distance  apart,  width, 


Depth  under  5  Fathoms.         5  to  10  Fathoms. 


10  to  20  Fathoms 


Over  20  Fatlioma. 


5  Miles. 


FIG.  294. 


etc.  Fig.  294  is  a  chart  of  the  Isle  of  Wight  and  the  surrounding  water, 
with  the  depths  expressed  as  shown  at  the  bottom  of  the  cut.  Sections  are  often 
used  for  rivers,  especially  for  those  like  our  Western  ones,  that  have  a  very 
changeable  bottom.  By  plotting  sections,  taken  at  different  times,  over  one 
another,  distinguishing  them  apart  by  a  difference  in  color  and  variety  of  line, 


160 


TOPOGRAPHICAL  DRAWING. 


the  changes  that  take  place  in  the  bottom  of  the  river,  and  the  erosion  of  the 
banks,  are  more  boldly  shown  than  by  the  use  of  any  other  method.  The 
ordinary  marine  conventionalities  are  as  follows  : 


DIRECTION  OF  THE  CURRENT 


Anchorage  for  ships, 
Anchorage  for  coasters,    j^ 
Rocks  always  covered,       j^ 


Buoys,  1 1 
Wrecks,  •£, 
Harbors, 


Light-house, 

Signal-house, 

Channel-marks, 


Representation  of  Geological  and  Statistical  Features. — The  geological  feat- 
ures of  a  country  may  be  readily  expressed  on  a  map  by  the  use  of  lines  as  in 


W.oPGr 


1.  Alluvia.  2.  Upper  Tertiary.        3.  London  Clay,  &c.  4.  Chalk.       5  &  6.  Greensand  and  Gait, 

10  &  11.  Triassic,  &c. 


16.  Silurian. 


12.  Permian. 


13.  Carboniferous. 


FIG.  295. 


marine  charts.     Fig.  295  is  a  geological  map  of  Southeastern  England,  and 
will  be  easily  understood  by  inspection. 


TOPOGRAPHICAL  DRAWING. 


161 


A  geological  profile  may  be  represented  in  the  same  way.  The  different 
rocks  or  formations  are  usually  distinguished  by  color  and  explained  by  mar- 
ginal notes  and  squares,  but  more  often  by  marks,  dots,  or  cross-hatchings,  as 


FIG.  296. 


in  Fig.  296,  which  exhibits  the  geological  features  of  the  United  States  east  of 
the  Rocky  Mountains  and  Canada  to  the  south  of  the  St.  Lawrence. 

Fig.  297  is  a  section  from  Pennsylvania  to  Canada,  showing  the  relations  of 
the  subdivisions  to  each  other. 
11 


162 


TOPOGRAPHICAL  DRAWING. 


Fig.  298  represents  an  ideal  diagram  of  the  principal  groups  in  American 
geology,  in  the  order  of  their  superposition. 


Ideal  Section  north  and  south  from  Canada  to  Pennsylvania :  A,  Archaean ;  L  S  and  U  S,  Silurian  ;  D,  De« 

vonian  ;  C1,  Carboniferous. 


ERAS. 
3.  P&ychozoic. 


4.  Cenozoic — 


3.  Mesozoic  ..< 


3.  Palaeozoic .  .< 


1.  Archaean,.. 


Carboniferous. 


Huronian. 


Lanrentian. 


FIG.  298. 

Ideal  General  Section  of  the  Whole  Series  of  Strata, 
stowing  the  Principal  Divisions  and  Subdivisions. 


Still  another  form  of  a  topographical 
and  statistical  map  is  shown  in  Fig.  299, 
which  is  a  portion  of  the  city  of  Lon- 
don, taken  from  a  sanitary  report  by 
a  commission  of  Parliament ;  and  em- 
bodies, in  a  graphic  way,  the  details  in 
regard  to  drainages,  natural  and  arti- 
ficial, contour-lines  and  street-sewers ; 
position  of  gas  and  water  mains,  and 
occupancy  of  buildings.  On  the  origi- 
nal are  also  given  the  number  of  the 
houses  and  names  of  streets. 

Eeference  has  been  made  to  the 
drawing  of  hills  by  contours,  and  it  has 
not  been  recommended  except  when  the 
lines  have  been  accurately  determined 
by  level.  When  this  is  the  case,  they 
should  always  be  used  ;  it  is  the  sim- 
plest and  most  explanatory  record  of 
facts,  and  if  the  facts  have  been  worth 
determining  they  are  worth  recording. 
When  contour-lines  are  brought  more 
closely  together  (as  shown  in  Fig.  300, 
which  is  from  the  same  sanitary  report, 
and  of  a  larger  portion  of  London),  it 
produces  the  effect  of  physical  relief, 
and  shows  at  a  glance  the  lines  of  natu- 
ral drainage,  and  from  it  profiles  can 
be  made,  in  any  direction,  for  the  grad- 
ing of  streets  or  sewers.  Were  town 
and  county  maps  thus  drawn  with  con- 
tour-lines, much  time  and  money  would 
be  saved  in  the  location  of  highways  and 
railways. 

Transferring. — It  is  usual,  in  plot- 
ting from  a  field-book,  to  make  first  but 
a  rough  draft,  and  then  make  a  finished 
copy  on  another  sheet.  In  the  first, 
many  lines  of  construction,  balances  of 


TOPOGRAPHICAL  DRAWING.  163 

survey,  and  trial  lines  are  drawn,  which  are  unnecessary  in  the  copy  ;  outlines 
of  natural  features  are  sketched  roughly,  but  the  plotting  of  surveys,  and  such 
lines  and  points  as  are  to  be  preserved  in  the  copy,  must  be  done  with  accuracy. 


FIG.  299. 
Private  houses  (occupied  by  persons  not  in  receipt  of  wages). 

Offices  and  shops. 

Houses  occupied  by  persons  in  receipt  of  wages. 

Warehouses. 

Stables  and  outhouses. 

Public  buildings. 
•     Contours  ;  vertical  distances  between  lines,  two  feet. 
•—•— =     Sewers. 
— — — — —     Gas-pipes. 
_,__i_5_i-    Water-pipes. 

The  most  common  way  of  transferring,  for  a  fair  copy,  is  by  superposition 
of  the  plan  above  the  sheet  intended  for  the  copy,  and  pricking  through  every 
intersection  of  lines  on  the  plan,  and  all  such  points  as  may  be  necessary  to 
preserve.  The  clean  paper  should  be  laid  and  fastened  smoothly  on  the  draw- 
ing-board ;  the  rough  draft  should  be  laid  on  smoothly,  and  retained  in  its 


164 


TOPOGRAPHICAL  DRAWING. 


position  by  weights,  glue,  or  tacks.  The  needle  must  be  held  perpendicular 
to  the  surface  of  the  plan,  and  pressed  through  both  sheets  ;  begin  at  one  side 
and  work  with  system,  so  as  not  to  prick  through  each  point  but  once,  nor 
omit  any ;  make  the  important  points  a  trifle  the  larger.  For  the  irregular 


FIG.  300. 

curves,  as  of  rivers,  make  frequent  points,  but  very  small  ones.  On  removing 
the  plan,  select  the  important  points,  those  defining  leading  lines ;  draw  in 
these,  and  the  other  points  will  be  easily  recognized  from  their  relative  position 
to  these  lines.  When  any  point  has  not  been  pricked  through,  its  place  may 
be  determined  by  taking  any  two  established  points  adjacent  to  the  one  re- 
quired, and  with  radii  equal  to  their  distance,  on  the  plan,  from  the  point 
required,  describing  arcs,  on  the  copy,  on  the  same  side  of  the  two  points  ;  their 
intersection  will  be  the  point  desired.  In  this  way,  as  in  a  trigonometrical 
survey,  having  established  the  two  extremes  of  a  base,  a  whole  plan  may  be 
copied.  In  extensive  drawings  it  is  very  common  to  prick  off  but  a  few  of  the 
salient  points,  and  fill  in  by  intersections,  as  above,  or  by  copying  detached 
portions  on  tracing-paper,  and  transferring  them  to  the  copy ;  the  position  of 
each  sketch  being  determined  by  the  points  pricked  off,  the  transfer  is  made 
by  pricking  through  as  above,  or  by  transfer-paper  placed  between  the  tracing 
and  the  copy. 

If  tracing  paper  or  cloth  (pages  56,  57)  be  placed  above  the  drawing,  every 
line  will  show  through,  and  can  be  traced  directly  with  the  pen,  in  India  ink. 
These  tracings  are  used  mostly  to  preserve  duplicates  of  finished  drawings. 

Duplicates  of  drawings,  contracts,  estimates,  etc.,  on  paper  allowing  the 
light  to  pass  through  are  readily  made  by  the  use  of  .ferro-prussiate  paper,  or 
the  blue-print  process.  Paper  can  be  prepared  by  washing  it  with  a  mixture 


TOPOGRAPHICAL  DRAWING.  165 

of  1£  ounces  of  citrate  of  iron  and  ammonia  with  8  ounces  of  water,  and  1£ 
ounces  of  red  prussiate  of  potash  and  8  ounces  of  water,  dissolved  separately 
and  mixed.  The  mixture  and  prepared  paper  should  be  kept  from  the  light. 
The  prepared  paper  in  close  rolls  can  be  readily  purchased.  For  the  manipu- 
lation there  is  needed  plate-glass,  and  a  blanket  a  little  larger  than  the  draw- 
ing, a  shallow  tin  dish,  that  the  drawing  can  be  placed  in  flat  for  washing. 
Lay  down  the  blanket  on  a  drawing-board,  above  that  the  ferro-prussiate 
paper,  next  the  drawing,  and  then  the  glass.  Expose  to  the  sunlight  for  about 
ten  minutes  if  the  drawing  is  on  tracing-paper  or  cloth,  and  longer  for  thicker 
paper ;  when  done,  the  background  should  be  a  metallic  gray.  Now  lay  the 
ferro-prussiate  paper  in  the  tin  dish,  cover  with  water,  and  leave  it  for  five  to 
ten  minutes  ;  wash  thoroughly  and  dry.  The  lines  will  be  white  on  a  blue 
ground.  The  negative  of  ferro-prussiate  paper  gives  blue  lines  on  a  white 
ground,  and  other  processes  black  lines  on  a  buff  ground. 

An  accurate  and  rapid  way  of  tracing,  on  drawing-paper,  plans  of  small 
extent,  is  by  means  of  an  instrument  called  a  copying-glass.  It  consists  of  a 
large  piece  of  plate-glass  set  in  a  frame  of  wood,  which  can  be  inclined  at  any 
angle.  On  this  glass  is  first  laid  the  original  plan,  and  above,  the  fair  sheet, 
and  the  frame  being  raised  to  a  suitable  angle,  a  strong  light  is  thrown  by  re- 
flectors or  otherwise  on  the  under  side  of  the  glass,  whereby  every  line  in  the 
original  plan  is  seen  distinctly  through  the  fair  sheet,  and  the  copy  is  made 
at  once,  as  on  tracing-paper.  This  same  process,  on  a  small  scale,  is  adopted 
by  putting  the  plans  upon  a  pane  of  glass  in  a  window. 

Plans  mounted  on  cloth,  or  on  opaque  paper,  do  not  admit  of  being  traced 
in  this  way.  In  such  cases  the  copy  may  be  made  by  means  of  transfer-paper. 
The  plan  is  first  traced  on  tracing-paper  or  cloth,  black-leaded  or  transfer  paper 
is  then  placed  on  the  fair  sheet,  and  the  tracing-paper  copy  is  placed  above. 
All  is  steadied  by  numerous  weights  along  the  edges,  or  by  drawing-pins  fixed 
into  the  drawing-board.  A  fine  and  smooth  point  is  then  passed  over  each 
boundary  or  mark  on  the  tracing  with  a  pressure  of  the  hand  sufficient  to 
cause  a  clear,  penciled  mark  to  be  left  on  the  fair  sheet  by  the  black-leaded 
or  transfer  paper.  The  whole  outline  is  thus  obtained,  and  afterward  drawn 
in  ink.  The  copyist  should  be  careful  in  his  manipulations,  so  as  not  to 
transfer  any  other  lines  than  those  required,  nor  leave  smutches  on  the  fair 
sheet. 

Plans  may  be  copied,  on  a  reduced  or  enlarged  scale,  by  means  of  the  pan- 
tagraph  (Figs.  146,  147),  or  by  the  method  of  squares  (pages  63,  64). 

Map  Projections. — For  a  farm  or  other  small  survey,  the  surface  of  the  earth 
can  be  conceived  to  be  flat,  and  the  map  a  horizontal  projection  of  the  plane 
surface  on  a  reduced  scale  ;  the  error  being  practically  insignificant,  while 
the  labor  is  greatly  reduced  by  making  this  assumption.  But,  for  large  maps 
of  countries,  States,  rivers,  etc.,  where  the  meridians  and  parallels  of  latitude 
are  represented,  such  a  system  would  be  so  erroneous  as  to  be  impracticable. 
The  surface  of  the  earth  being  a  sphere,  it  is  incapable  of  development  on  a 
plane,  so  that  it  becomes  necessary  to  make  the  best  approximation  possible  in 
form,  relation,  and  proportional  area  of  the  portions  to  be  represented  on  a 
map  or  chart.  There  are  many  different  kinds  of  projection,  all  more  or  less 


166 


TOPOGRAPHICAL  DRAWING. 


imperfect,  but  most  of  which  possess  advantages  for  some  descriptions  of  maps 
or  charts.     They  may  be  divided  into  four  classes,  as  follows  : 

Class      I.  Perspective  projection  on  planes. 
"      II.  Developed  perspective  projections. 
"    III.  Projections  by  developing  elements. 
"     IV.  Projections  conformed  to  some  arbitrary  condition. 

In  Class  I,  the  more  important  kinds  are  the  globular  or  equidistant,  and 
the  stereographic. 

Globular  or  Equidistant  Projection  of  the  Sphere.— According  to  this 
method  the  eye  is  placed  at  a  distance  from  the  center  of  the  earth,  equal  to 
1.707  x  radius.  The  plane  of  projection  passes  through  the  center  perpen- 
dicular to  the  central  ray.  This  method  is  quite  common  in  school  maps. 
The  following  is  the  construction  : 

Draw  two  lines  (Fig.  301),  at  right  angles  to  and  intersecting  each  other  ; 
from  the  point  0  of  their  intersection  as  a  center,  with  a  radius  equal  to  that 


FIG.  301. 

intended  for  the  hemisphere,  describe  a  circle,  and  mark  the  points  N",  S,  W,  E. 
N  and  S  will  be  the  poles,  the  line  N  S  the  central  meridian,  and  W  E  the 
equator.  Divide  N  S  and  W  E  into  as  many  equal  parts  as  there  are  degrees 
or  numbers  of  degrees  to  be  represented — in  the  figure  in  divisions  of  30° — and 
meridian  and  equator  into  six  equal  parts,  as  the  hemisphere  embraces  180°. 
Commence  at  C,  and  divide  the  half-lines  into  three  equal  parts.  Divide  the 
arcs  N  W,  N  E,  S  W,  and  S  E,  each  into  three  equal  parts.  There  will  be  now 
determined  three  points  in  two  parallels  of  north  and  south  latitude,  30°  and 
60°,  through  which  to  describe  the  arcs  representing  the  parallels.  The  center 
of  these  arcs  will  be  in  the  line  N  S  ;  describe  the  arc,  and  with  the  same  radius 
from  a  center  on  the  line  N  S  below  the  S  pole,  describe  a  similar  arc  passing 
through  the  S  30°  point  on  the  meridian.  Therefore,  keeping  the  steel  point 
of  the  dividers  on  the  line  N  S,  by  trial  radii  may  be  found  of  arcs  which  shall 


TOPOGRAPHICAL  DRAWING. 


167 


pass  through  the  points  on  the  central  meridian  and  on  the  circle.  With  the 
radii  describe  arcs  for  the  parallels  in  north  and  south  latitude.  All  the  me- 
ridians pass  through  the  N  and  S  poles,  and  through  the  divisions  of  degrees  on 
the  equator.  There  are  three  points,  therefore,  determined  in  the  arc  of  each 
meridian  which  may  be  described  from  centers  found  by  trial  on  the  line  E  W. 
Stereographs  Projection. — In  this  method  the  eye  is  taken  at  the  center  of 
the  earth,  at  the  pole  of  the  great  circle  used  as  a  plane  of  projection.  Circles 
are  stereographically  projected  into  circles.  An  increasing  exaggeration  out- 
ward from  the  center  is  its  principal  defect.  To  project  stereographically  the 
hemisphere  on  the  plane  of  the  meridian,  draw  the  central  meridian,  equator, 


FIG.  303. 

and  circle  (Fig.  302),  as  in  the  preceding  problem.  To  project  the  other  me- 
ridians (say  every  10°),  divide  the  quadrant  N  E  into  nine  equal  parts  ;  from 
S  to  these  points  of  division,  10,  20,  30,  draw  lines  intersecting  C  E  in  10,  20, 
30.  These  latter  points  are  in  the  meridians  through  which  N  and  S  arcs  are 
to  be  described  from  centers  on  the  line  E  W. 

To  find  in  like  manner  the  three  points  in  the  parallels  of  latitude,  divide 
the  quadrants  into  nine  parts,  80,  70,,  60,  and  through  these  points  draw  lines 
to  W  ;  the  intersection?  with  the  central  meridian  80,  70,  60,  will  with  the 
points  of  the  quadrant  furnish  three  points  through  which  to  describe  arcs  of 
parallels  of  latitude. 

To  project  the  hemisphere  on  the  plane  of  the  equator  (Fig.  303).  Draw  two 
lines  at  right  angles  to  each  other  ;  describe  the  circle  and  divide  the  circum- 
ference as  before.  The  center  0  will  be  the  projection  of  N  or  S  pole,  the  lines 
at  right  angles  to  each  other  will  be  meridians,  as  well  as  any  other  diameters, 
as  D  H,  F  K,  drawn  through  some  division  of  the  circumference. 

To  project  the  parallels  of  latitude.  The  circle  represents  the  projection  of 
the  equator,  and  the  other  parallels  must  be  arcs  on  the  same  center  C,  of 
which  the  radii  are  to  be  determined  by  the  intersections  of  the  line  C  B  by 
lines  drawn  from  A  to  the  divisions  of  the  circle  10,  20,  30. 

In  Class  II,  instead  of  projecting  directly  on  planes,  an  intermediate  cone 
or  cylinder  is  employed  to  receive  the  projection,  which  is  then  developed  on  a 


168 


TOPOGRAPHICAL  DRAWING. 


tangent  plane.  The  cylinder  or  cone  must  always  be  employed,  because  they 
are  the  only  surfaces  that  can  be  developed  on  a  plane.  The  eye  is  always  con- 
ceived to  be  at  the  center  of  the  earth  in  all  the  projections  of  this  class. 

In  Class  III  the  portions  of  the  earth's  surface  are  mapped  by  being  divided 
into  small  or  differential  elements  which  are  successively  developed.  This 
method  admits  of  greater  accuracy  than  any  of  the  four  classes.  The  two  most 
important  subdivisions  are  Bonne's  and  the  Polyconic. 

In  Bonne's  projection,  assume  a  central  meridian,  and  a  central  parallel 
with  a  cone  tangent  along  the  latter.  The  central  meridian  is  then  developed 
on  that  element  of  this  cone  to  which  it  is  tangent,  and  the  cone  is  then  de- 
veloped on  a  tangent  plane.  The  parallel,  by  this  process,  becomes  an  arc 
with  its  center  at  the  vertex  of  the  cone,  and  the  meridian  becomes  a  graduated 
line.  Conceive  concentric  circles  to  be  traced  through  points  on  this  meridian 
at  elementary  distances  apart.  The  zones  of  the  sphere  situated  between  the 
parallels  through  these  points  are  then  conceived  to  be  developed  each  between 
its  corresponding  arcs.  In  this  way  all  the  i:ones  of  the  sphere  are  developed 
on  a  plane  surface  in  their  true  relation  to  each  other  and  the  central,  each 
having  the  same  length,  width,  and  relation  to  its  neighboring  zone  that  it  did 
on  the  spherical  surface.  The  areas  are  not  changed  by  the  development,  and 
distances  along  the  parallels  are  correct,  while  those  along  the  meridians  are 
slightly  increased,  except  those  along  the  central  meridian,  which  are  strictly 
correct.  The  scale  is  nearly  uniform  over  the  whole  map,  and,  for  moderate 
areas,  the  intersections  are  nearly  rectangular.  Bonne's  method  is  almost 
universally  applied  to  the  detailed  topographical  maps  based  on  the  trigonomet- 
rical surveys  of  the  different  states  of  Europe. 

The  Polyconic  has  been  adopted  by  the  United  States  Coast  Survey,  and 
all  their  maps  are  projected  by  this  method.  Each  parallel  is  supposed  to  be 
represented  on  a  plane  by  the  development  of  a  cone  having  the  parallel  for  its 
base,  and  its  vertex  at  the  point  of  intersection  of  a  tangent  to  the  parallel  and 
the  earth's  axis.  The  map  thus  becomes  the  development  of  the  surface  of 
successive  cones,  and  the  degrees  of  the  parallel  preserve  their  true  length. 
The  following  tables  are  given  for  use  in  projecting  large  maps.  Their  use 
will  be  explained  in  an  example.  For  making  small  maps,  with  a  great  de- 
gree of  accuracy,  tables  are  published  by  the  United  States  Coast  Survey. 

Co-ordinates  of  Curvature  in  Miles  for  Maps  of  Large  Extent. 


Latitude  20°. 

Latitude  24". 

Latitude  28°. 

Latitude  32°. 

LONGITUDE. 

D.  M. 

D.  P. 

D.  M. 

D.  P. 

D.  M.                   D.  P. 

D.  M. 

D.  P. 

2. 

130-0 

0-8 

126-4 

0-9 

122-2 

1-0 

117-4 

M 

4. 

260-0 

3-1 

252-8 

3-6 

244-4 

4-0 

234-8 

4-3 

6. 

390-0 

6-9 

379-2 

8-1 

366-5 

9-0 

352-0 

9-8 

8. 

620-0 

12-4 

505-5 

14-4 

488-6 

16-0 

46S-3 

17-3 

10. 

649-8 

19-4 

631-7 

22-4 

610.4 

25-0 

586-3 

27-1 

12. 

779-7 

27-8 

757-9 

32-2 

732-4 

36-0 

703-5 

39-1 

14 

909-2 

38-0 

883-6 

43-9 

853*7 

49-0 

819-6 

53-1 

16. 

1039-2 

49-6 

1009-9 

57-4 

975-7 

64-1 

936-8 

69-5 

18. 

1168-1 

62-8 

1134-8 

726 

1096-0 

80-9 

1051-9 

87.8 

20. 

1298-0 

77-6 

1261-2 

89-7 

1218-8 

100-1 

1169-2 

108-6 

R. 

10892 

8905 

7458 

6348 

TOPOGRAPHICAL  DRAWING.  169 

Co-ordinates  of  Curvature  in  Miles  for  Maps  of  Large  Extent. — (Continued.) 


Latitude  36°. 

Latitude  40°. 

Latitude  44°. 

Latitude  48°. 

LONGITUDE. 

I).  M. 

D.  P. 

D.  M. 

D.  P. 

D.  M. 

D.  P. 

D.  M. 

D.  P. 

2     . 

112-0 

1-2 

106-1 

1-2 

99-7 

1-2 

92-7 

1-2 

4.  . 

224-0 

4-6             212-2 

4-8 

198-9 

4-8 

185-4 

4-8 

6.  . 

335-9 

10-3 

318-1 

10-7 

298-7 

10-9 

277-9 

10-8 

8.. 

447-7 

18-4 

423-9 

18-9 

398-0 

19-3 

370-3 

19-2 

10.. 

659-2 

28-7 

529-4 

29-7 

497-1 

30-2 

462-3 

30-0 

12.. 

670-5 

41-3 

634-7 

42-8 

595-9 

48-4 

654-1 

43-2 

14.  . 

781-6 

56-2 

739-7 

58-2 

694-3 

59-1 

645-6 

68-8 

16.. 

892-3 

73-4 

844-3 

76-0 

792-3 

77'1 

736-5 

76-7 

18.. 

1002-6 

92-8 

948-5 

96-1 

889-9 

97-5 

827-0 

97-0 

20.. 

1112-5 

114-5 

1052-3 

1185 

986-9 

120-2 

916-9 

119.6 

R. 

5461 

4729 

4110 

3575 

Length  of  a  Degree  of  Longitude  at  Different  Latitudes,  and  at  Sea-Level. 


Deg. 
of 
Lat. 

Miles. 

»„?• 

Lat. 

Miles. 

Deg. 
of 
Lat. 

Miles. 

"of' 
Lat. 

Miles. 

Deg. 
of 
Lat. 

Miles. 

Deg. 
of 
Lat. 

Miles. 

0 

69-16 

14 

67-12 

28 

61-11 

;   42 

51-47 

56 

38-76 

70  1       23-72 

2 

69-12 

16 

66-50 

30 

59-94 

44 

49-83 

58 

36-74 

72 

21-43 

4 

68-99 

18 

65-80 

32 

58-70 

46 

48-12 

60 

34-67 

74 

19-12 

6 

68-78 

20 

65-02 

34 

57-39 

48 

46-36 

62 

32-55 

76 

16-78 

8 

68-49 

22 

64-15 

36 

56-01 

50 

44-54 

64 

30-40 

78 

14-42 

10 

68-12 

24 

63-21 

38 

54-56 

52 

42-67 

66 

28-21 

80 

12-05 

12 

67-66 

26 

62-20 

40 

53-05 

54 

40-74 

68 

25-98 

82 

9-66 

Lengths  for  intermediate  degrees  can  be  found  accurately  by  proportion. 
At  the  equator,  1°  of  latitude  =  68 '70  miles  ;  at  latitude  20°  =  68 '78  ;  at  40° 
=  69-00  ;  at  60°  =  69-23  ;  at  80°  =  69-39  ;  at  90°  =  69-41. 

To  draw  a  map  according  to  the  tables,  we  lay  off  on  the  straight  line  (Fig. 
304)  N"  S,  representing  the  middle  meridian,  the  lengths  representing  the  ten 
degrees  of  latitude  between  20°  and  30°,  30°  and  40°,  etc.  Through  these 
points  draw  circular  arcs  with  the  radii  designated  by  R  in  the  preceding  tables. 
On  these  arcs  lay  off  the  lengths  of  ten  degrees  of  longitude  for  each  correspond- 
ing latitude  on  each  side  of  the  center  meridian.  Through  the  points  thus 
formed  draw  the  meridians,  which  will  be  found  slightly  concave  toward  the 
middle  one.  If  the  scale  is  so  large  that  it  is  impossible  to  draw  the  circular 
arcs  with  beam-compasses,  erect  perpendiculars  at  the  points  20°,  30°,  40°,  and 
50°,  and  on  them  lay  off  the  values  d  m  from  the  tables.  At  each  of  the  points 
so  found  erect  perpendiculars,  and  set  off  on  them  the  corresponding  values  of 
d  p.  Through  the  points  thus  found  draw  the  parallels  and  meridians.  The 
principal  advantages  of  this  projection  are — a  minimum  amount  of  distortion 
at  any  portion  of  the  map  ;  a  scale  of  degrees  and  minutes  of  the  parallels 
and  meridians,  by  means  of  which,  positions,  determined  by  their  latitudes  and 
longitudes,  may  be  readily  inserted  on  the  maps  ;  the  use  of  a  linear  scale  in 
any  portion  or  direction  of  the  map ;  and  the  intersection  of  parallels  and 
meridians  at  nearly  right  angles. 

In  Class  /Fsome  arbitrary  mathematical  condition  is  imposed,  for  some 
practical  purpose,  usually  giving  rise  to  distorted  maps. 


iro 


TOPOGRAPHICAL  DRAWING. 


For  polar  projections,  De  Lorgne's  has  much  merit.     Calculate  first  a  cir- 
cle with  an  area  equivalent  to  that  of  the  hemisphere  to  be  projected.     Draw 

N 


such  a  circle  and  connect  the  graduations  of  the  circumference  with  the  center. 
These  represent  the  meridians.     The  radius  can  be  divided  into  ninety  equal 


N 


W 


80 

eo 

<.o 

20 

160 

/40 

120 

100 

80 

60 

W 

20 

20 

p 

20 

VO 

60 

80 

Too 

120 

/to 

160 

.0 

60 

80 

s 

FIG.  305. 


parts  ;  but,  where  it  is  possible,  the  chords  of  the  polar  distances  of  the  par- 
allels should  be  used  for  determining  the  parallels. 


TOPOGRAPHICAL  DRAWING. 


m 


Mercator's  chart  is  especially  valuable  to  the  navigator.  By  it  he  can  lay 
off  his  course  accurately  on  the  chart  in  a  straight  line.  It  has  little  value  for 
the  other  purposes  of  a  chart.  Meridians  are  represented  by  equidistant  par- 
allel straight  lines,  and  the  parallels  by  a  perpendicular  set  of  parallel  straight 
lines,  whose  distances  from  each  other  increase  from  the  equator  outward  in 
the  same  ratio  as  the  corresponding  longitudinal  degrees  diminish.  By  this 
means  the  relation  between  the  latitude  and  longitude  measurements  on  the 
chart  is  preserved  uniformly  as  on  the  earth's  surface. 

To  construct  a  Mercator's  Chart  (Fig.  305).  Draw  two  straight  lines,  W  E 
and  N  S,  intersecting  each  other  at  right  angles  at  C.  W  E  is  the  equator,  N  S 
the  meridian  passing  through  the  middle  of  the  chart.  From  0  set  off  equal 
parts  on  the  equator  both  ways,  to  represent  degrees  of  longitude,  subdivided 
into  minutes  if  the  size  of  the  chart  will  admit  of  it.  Assuming  the  equator 
as  a  scale  of  minutes,  set  off  from  C,  toward  N  and  S,  the  number  of  minutes  in 
the  enlarged  meridian  corresponding  to  each  degree  of  latitude,  as  shown  by 
the  table  of  meridional  parts.  Draw  lines  parallel  to  N  S  through  the  divisions 
of  the  equator  for  meridians,  and  parallels  to  W  E  through  the  divisions  of 
N  S  for  parallels  of  latitude. 

To  find  the  bearing  of  any  one  place  from  another,  it  is  only  necessary  to 
draw  a  straight  line  between  the  two  points,  and  observe  the  angle  it  makes 
with  the  meridians. 

Table  of  Meridional  Parts. 


Latitude. 

Meridional  parts. 

Latitude. 

Meridional  parts. 

Latitude. 

Meridional  parts. 

0 

0 

0 

0 

o-oo 

35 

2244-29 

70 

5965-92 

5 

300-38 

40 

2629-69 

75 

6970-34 

10 

603-07 

45 

3029-94 

80 

8375-20 

15 

910-46 

50 

3474-47 

85 

10764-62 

20 

1225-14 

55 

3967-97 

90 

Infinite. 

25 

1549-99 

60 

4527-37 

30 

1888-38 

65 

5178-81 

COLORED   TOPOGRAPHY. 

Topographical  features  may  be  represented  effectively  and  expeditiously  by 
means  of  the  brush  and  water-colors,  either  by  India  ink  alone,  or  by  various 
tints,  or  by  the  union  of  both. 

The  most  important  colors  for  conventional  tints  are  (besides  India  ink), 
indigo  (blue),  carmine  (or  crimson  lake),  and  gamboge  (yellow),  used  separately 
or  compounded.  Besides  these,  burnt  sienna,  yellow  ochre,  and  vermilion  are 
sometimes  used,  although  the  first  three  are  susceptible  of  the  best  combina- 
tions, and  the  others  are  generally  used  alone. 

The  following  conventional  colors  are  used  by  the  French  military  engineers 
in  their  colored  topography  :  Woods,  yellow  ;  using  gamboge  and  a  very  little 
indigo.  Grass-land,  green  ;  made  of  gamboge  and  indigo.  Cultivated  land, 
brown  ;  lake,  gamboge,  and  a  little  India  ink  ;  "burnt  sienna"  will  answer. 
Adjoining  fields  should  be  slightly  varied  in  tint.  Sometimes  furrows  are  in- 
dicated by  strips  of  various  colors.  Gardens  are  represented  by  small  rectangu- 


172  TOPOGRAPHICAL  DRAWING. 

lar  patches  of  brighter  green  and  brown.  Uncultivated  land,  marbled  green 
and  light  brown.  Brush,  brambles,  etc.,  marbled  green  and  yellow.  Heath, 
furze,  etc.,  marbled  green  and  pink.  Vineyards,  purple ;  lake  and  indigo. 
Sands,  a  light  brown;  gamboge  and  lake;  "yellow  ochre  "will  do.  Lakes 
and  rivers,  light  blue,  with  a  darker  tint  on  their  upper  and  left-hand  sides. 
Seas,  dark  blue,  with  a  little  yellow  added.  Marshes,  the  blue  of  water,  with 
spots  of  grass  green,  the  touches  all  lying  horizontally.  Eoads,  brown  ;  be- 
tween the  tints  for  sand  and  cultivated  ground,  with  more  India  ink.  Hills, 
greenish  brown  ;  gamboge,  indigo,  lake,  and  India  ink.  Woods  may  be  finished 
Up  by  drawing  the  trees  and  coloring  them  green,  with  touches  of  gamboge 
toward  the  light  (the  upper  and  left-hand  side),  and  of  indigo  on  the  opposite* 
side. 

In  addition  to  the  conventional  colors,  a  sort  of  imitation  of  the  conven- 
tional signs  is  introduced  in  color  with  the  brush,  and  shadows  are  almost 
invariably  introduced.  The  light  is  supposed  to  come  from  the  upper  left-hand 
corner,  and  to  fall  nearly  vertical,  but  sufficiently  oblique  to  allow  of  a  decided 
light  and  shade  to  the  slopes  of  hills,  trees,  etc.  After  the  shadow  has  been 
painted,  the  outline  of  the  object  is  strengthened  by  a  heavy  black  line  on 
the  side  opposite  the  light.  The  flat  tints  are  first  laid  on  as  above,  and 
then  the  conventional  signs  are  drawn  in  with  a  pencil  and  colored  in  with 
appropriate  and  more  intense  tints ;  the  shadows  are  generally  represented  in 
India  ink. 

Hills  are  usually  shaded,  not  as  they  would  appear  in  nature,  but  on  the 
conventional  system  of  making  the  slopes  darker  in  proportion  to  their  steep- 
ness ;  the  summits  of  the  highest  ranges  being  left  white — an  arrangement 
incorrect  in  theory,  but  generally  understood  by  those  not  accustomed  to  plan- 
drawing,  and  is  easy  of  execution.  Wash  the  surface  first  with  the  proper  flat 
tint,  trace  in  with  a  pencil  outlines,  then  lay  on  in  India  ink  tints  propor- 
tioned in  intensity  to  the  height  of  the  hills  and  steepness  of  the  slopes.  To 
soften  the  tints,  two  brushes  are  used,  one  as  a  color-brush,  the  other  as  a  water- 
brush  :  the  tints  are  laid  on  with  the  first,  and  softened  by  passing  the  water- 
brush  rapidly  along  the  edges.  The  water-brush  must  not  have  too  much 
water,  as  it  would  in  that  case  lighten  the  tint  to  a  greater  extent  than  is  in- 
tended, and  leave  a  ragged,  harsh  edge.  Tints  may  be  applied  in  very  light 
shades,  one  tint  over  another,  with  the  boundary  of  the  upper  tint  not  reaching 
the  extreme  limit  of  the  tint  below  it.  When  depth  of  shade  is  required,  it  is 
best  produced  by  application  of  several  light  tints  in  succession  ;  no  tint  is  to 
be  laid  over  the  other  until  the  first  is  dry. 

When  woods  have  to  be  represented,  the  shading  used  for  the  trees,  instead 
of  interfering  with  the  shadows  due  to  the  slopes,  may  be  made  to  harmonize 
with  them,  and  contribute  to  the  general  effect  by  presenting  greater  or  less 
depth,  according  to  the  position  of  the  woods  on  the  sides,  or  summits  of  the 
hills. 

An  expeditious  and  effective  way  of  representing  hills  with  a  brush,  a  spe- 
cies of  imitation  of  hills  drawn  with  a  pen  on  the  vertical  system,  is  effected  by 
pressing  out  flat  the  brush  to  a  sort  of  comb-like  edge  ;  drawing  this  over  a 
nearly  dry  surface  of  India  ink,  and  then  brushing  lightly  or  more  heavily  be- 


TOPOGRAPHICAL   DRAWING.  173 

tween  the  contours,  according  to  the  steepness  of  the  slope,  each  of  the  comb- 
like  teeth  making  its  mark. 

Kivers  and  masses  of  water  may  be  shaded  in  with  a  color  and  water  brush 
as  above,  or,  by  superposition  of  light  tints,  a  shadow  may  be  thrown  from  the 
bank  toward  the  light,  and  the  outline  of  this  bank  strengthened  with  a  heavy 
black  line.  The  tints  are  to  be  in  indigo,  the  shadows  in  India  ink. 

Topographical  drawings  may  be  made  in  water-color  with  but  one  tint,  as 
India  ink,  or  ink  mixed  with  a  little  sepia.  The  conventional  signs  are  in 
imitation  of  pen-drawings,  the  hills  in  softened  tint,  or  drawn  with  the  comb- 
edged  brush,  and  the  rivers  shaded  with  superposed  tints. 

Most  artistic  and  effective  drawings  are  made  of  hills  as  they  would  appear 
in  nature,  under  an  oblique  light ;  the  sides  of  the  hills  next  the  light  receiving 
it  more  or  less  brilliantly,  according  as  they  are  inclined  more  or  less  at  right 
angles  with  its  rays,  and  the  shades  on  the  sides  removed  from  the  light,  increas- 
ing in  intensity  as  the  slopes  increase  in  steepness. 

Having  damp-stretched  the  paper  upon  the  drawing-board,  first  draw  in 
the  lines  in  pencil,  and  afterward  repeat  them  with  a  very  light  ink-line  ;  a 
soft  sponge,  well  saturated,  should  then  be  passed  quickly  over  the  surface  of 
the  drawing,  in  order  to  remove  any  portions  of  the  ink  which  would  be  liable 
to  mix  with  the  tint  and  mar  its  uniformity. 

The  moistened  surface  will  prevent  the  tint  from  drying  too  rapidly  at  the 
edges.  In  tinting,  "never  allow  the  edge  to  dry  until  the  whole  surface  is  cov- 
ered ;  leave  a  little  superfluous  color  along  the  edge  while  filling  the  brush.  In 
applying  a  flat  tint  to  large  surfaces,  let  the  drawing-board  be  inclined  upward 
at  an  angle  of  five  or  six  degrees,  so  as  to  allow  the  color  to  flow  downward  over 
the  surface.  With  a  moderately  full  brush,  commence  at  the  upper  outline, 
and  carry  the  color  along  uniformly  from  left  to  right  and  from  right  to  left  in 
horizontal  bands,  taking  care  not  to  overrun  the  outlines,  in  approaching  which 
the  point  of  the  brush  should  be  used,  and  at  the  lower  outline  let  there  be 
only  sufficient  color  in  the  brush  to  complete  the  tinting. 

No  color  should  be  allowed  to  accumulate  in  inequalities  of  the  paper,  but 
should  be  evenly  distributed  over  the  whole  surface. 

Too  much  care  can  not  be  given  to  the  first  application  of  color ;  as  any 
attempt  to  remedy  a  defect  by  washing  or  applying  fresh  tints  will  be  found 
extremely  difficult,  and  to  generally  make  bad  worse. 

Erasers  should  never  be  used  on  a  tinted  drawing,  as  the  paper,  when 
scratched,  receives  the  tint  more  readily,  and  retains  a  larger  portion  of  color 
than  other  parts,  thereby  causing  a  darker  tint. 

Marbling  is  done  by  using  two  separate  tints,  and  blending  them  at  their 
edges.  A  separate  brush  is  required  for  each  tint  ;  before  the  edge  of  the  first 
is  dry,  pass  the  second  tint  along  the  edge,  blending  one  tint  into  the  other, 
and  continue  with  each  tint  alternately. 

In  reference  to  the  general  effect  to  be  produced  in  tinted  topographical 
drawings,  as  to  intensity,  everything  should  be  subordinate  to  clearness ;  no 
tint  should  be  prominent  or  obtrusive.  Tints  that  are  of  small  extent  must  be 
a  little  more  intense  than  large  surfaces,  or  they  will  appear  lighter  in  shade. 
Keep  a  general  tone  throughout  the  whole  drawing.  Beginners  will  find  it  best 


174  TOPOGRAPHICAL   DRAWING. 

to  keep  rather  low  in  tone,  strengthening  their  tints  as  they  acquire  boldness 
of  touch. 

Plate  VIII  gives  an  example  of  colored  topography. 

In  lettering  tinted  drawings,  let  the  letters  harmonize  with  the  rest  of  the 
plan  ;  let  them  be  in  tint  more  intense  than  the  topography,  prominent  but 
not  obtrusive. 

Finishing  the  Plan  or  Map. — In  general,  in  topographical  drawings,  the 
light  is  supposed  to  fall  upon  the  surface  in  a  diagonal  direction  from  the 
upper  left-hand  corner.  This  rule  is  not  uniform  ;  by  some  draughtsmen  the 
light  is  introduced  at  the  lower  left,  and  hills  are  mostly  represented  under  a 
vertical  light,  although  the  oblique  adds  more  to  the  picturesque  effect.  The 
plan  is  usually  so  drawn  that  the  top  may  represent  the  north,  and  the  upper 
left-hand  corner  is  then  the  northwest. 

In  inking  in,  commence  first  with  the  light  lines,  since  a  mistake  in  these 
lines  may  be  covered  by  the  shade-lines.  Describe  all  curves  which  are  to  be 
drawn  with  compasses  or  sweeps  before  the  straight  lines,  for.it  is  easier  to  join 
neatly  a  straight  line  to  a  curve  than  the  opposite.  Ink  in  with  system,  com- 
mencing say  at  the  top  ;  ink  in  all  light  lines  running  easterly  and  westerly, 
then  all  light  lines  running  northerly  and  southerly,  then  commence  in  the 
same  way  and  draw  in  the  shade-lines.  It  will  of  course  be  understood  that 
elevated  objects  have  their  southern  and  eastern  outline  shaded,  while  depres- 
sions have  the  northern  and  western  ;  thus,  in  conventional  signs,  roads  are 
shaded  the  opposite  to  canals.  Having  inked  in  all  lines  that  are  drawn  with 
a  ruler  or  described  with  compasses,  commence  again  at  one  corner  to  fill  in 
the  detail,  keeping  all  the  rest  of  the  plan  except  what  you  are  actually  at  work 
upon  covered  with  paper,  to  protect  it  from  being  soiled.  The  curved  lines  of 
brooks,  fences,  etc.,  are  sometimes  drawn  with  a  drawing-pen,  sometimes  with 
a  steel  pen  or  goose-quill.  The  latter  are  generally  used  in  drawing  the  verti- 
cal lines  of  hills. 

Boundary-lines  of  private  properties,  of  townships,  of  counties,  of  States, 
etc.,  are  indicated  by  various  combinations  of  short  lines  and  dots,  thus  : 


All  plans  should  have  meridian  lines  drawn  on  them  ;  also  scales,  and  the 
dates  on  which  the  plans  were  finished.  Page  175  gives  several  designs  for 
meridians  and  borders.  In  these  diagrams  it  will  be  observed  that  both  true 
and  magnetic  meridians  are  drawn  ;  this  is  desirable  when  the  variation  is 
known,  but  in  many  surveys  merely  the  magnetic  meridian  is  taken ;  in  these 
cases  this  line  is  simply  represented  with  half  of  the  barb  of  the  arrow  at 
the  north  point,  and  on  the  opposite  side  of  the  line  from  the  true  meridian. 
Scales  are  drawn  or  represented  in  various  forms,  or  the  proportion  of  the  plan 
to  the  ground  is  expressed  decimally,  as  the  number  of  feet,  chains,  etc.,  to 
the  inch. 

Lettering. — The  style  in  which  this  is  done  very  much  affects  the  general 
appearance  of  the  plan.  Great  care  must  be  taken  in  the  selection  and  char- 
acter of  the  type,  and  in  the  execution. 


TOPOGRAPHICAL  DRAWING. 


175 


176  TOPOGRAPHICAL  DRAWING. 

MAP  OF 
EXPLORATIONS  AND    SURVEYS 

IN 

NEW  MEXICO  AND  UTAH 

made  under  the  direction  of  die 

SECRETARY  OF  WAR 

by 

CAPT.  J.N.MACOMB   TOP^ENG".8 

assisted  by 

C,H.DIMMOCK,  C.ENG* 
I860 

Scale  of  12  Miles  to  one  Inch  or  1:760320 
fc"-" — -4 — "  '  -*-i 53 20— x « "'        >o 


In  the  chapter  on  Drawing  Instruments  examples  of  the  method  of  con- 
structing letters,  as  well  as  some  alphabets,  are  given. 

Titles* — On  this  pageare  given  some  examples  of  titles,  intended  merely  as  an 
illustration  of  the  form  of  letters  and  their  arrangement,  the  scale  being  much 
smaller  than  that  used  on  plans,  except  such  as  are  drawn  to  a  small  scale.  It 


TOPOGRAPHICAL  DRAWING.  177 


will  be  observed  that  the  more  important  words  are  made  in  prominent  type. 
The  lower  part  of  the  title  should  always  contain,  in  small  character,  the  name 
of  the  party  making  the  survey,  and  also  the  name  of  the  draughtsman,  with 
date  of  the  execution  of  the  plan  :  if  the  survey  was  made  some  time  previous, 
the  date  of  the  survey  should  be  given.  If  the  plan  is  compiled  from  several 
surveys,  the  authorities  should,  if  possible,  be  given.  The  lettering  of  the  title 
in  lines  parallel  to  the  bottom  of  the  plan  is  preferable,  and,  in  general,  the 
great  mass  of  lettering  in  the  body  of  the  plan  should  be  formed  in  similar 
lines  ;  but  curved  lines  are  often  not  only  essential,  but  they  materially  con- 
tribute to  the  beauty  of  the  plan.,  Thus,  on  crooked  boundaries,  on  outlines 
of  maps,  the  lettering  should  follow  the  general  curve  of  the  boundary  ;  also 
on  crooked  rivers,  lakes,  seas,  etc.  ;  on  irregular  or  straggling  pieces  of  land, 
in  order  to  show  the  extent,  connection,  or  proprietorship  thereof,  the  lettering 
should  follow  the  central  line  of  such  a  tract  ;  and,  if  pieces  of  land  be  very 
oblong  in  form  but  regular  in  outline,  the  lettering  will  be  central  in  the 
direction  of  the  longest  side.  The  lettering  of  roads,  streets,  etc.,  is  always 
in  the  direction  of  the  line  of  road.  Curved  lines  of  lettering  are  often  intro- 
duced into  extended  titles  to  take  oif  the  monotonous  appearance  presented 
by  a  great  number  of  straight  lines  of  writing. 

The  direction  of  all  lettering  should  be  so  as  to  be  read  from  left  to  right. 
If  shades  or  shadows  are  introduced,  they  should  be  uniform  with  the  rest  of 
the  plan. 

It  will  be  observed  that  letters  vary  very  considerably  in  their  width,  the  / 
being  the  narrowest,  and  the  W  the  widest  ;  if,  therefore,  the  letters  composing 
a  word  be  spaced  off  at  equal  distances  from  center  to  center,  the  interval  or 
space  between  the  letters  will  be  more  in  some  cases  than  in  others.  Thus,  in 
the  word 

R   A    I    L  W  A  Y 

To  avoid  this,  write  in  first  one  letter,  and  then  space  off  a  proper  interval, 
and  then  write  in  the  next  letter,  and  then  space  off  the  interval  as  before, 
and  so  on,  thus  : 

RAILWAY 

When,  as  frequently  happens,  the  words  are  very  much  extended,  in  order  to 
embrace  and  explain  a  large  extent  of  surface  or  boundary,  and  the  space  occu- 
pied by  the  letter  is  small  in  comparison  with  the  interval,  the  disparity  of 
intervals  will  not  be  noticed,  and  the  letters  may  be  then  laid  off  at  equal 
spaces  from  center  to  center,  thus  : 

R        A         I          L        W        A        Y 

When  the  lines  of  lettering  are  curved,  the  same  rules  for  spacing  are  to  be 
observed  as  above.  If  the  letters  are  upright,  as  Roman  or  Gothic,  the  sides 
of  each  letter  are  to  be  parallel  to  the  radius  drawn  to  the  center  of  the  letter, 
and  the  bottom  and  top  lines  at  right  angles  to  it.  If  the  letters  be  inclined, 
12 


178 


TOPOGRAPHICAL  DRAWING. 


as  Italic  letters,  then  the  side-lines  of  the  letters  must  be  inclined  to  the  central 
radial  line,  as  on  a  horizontal  line  they  are  inclined  to  the  perpendicular. 


In  laying  off  letters  by  equal  intervals,  it  is  usual  to  count  the  number  of 
letters  in  the  word,  and  fix  the  position  on  the  plan  of  the  central  one,  and 
then  space  off  on  each  side  ;  this  is  particularly  important  in  titles,  when  it  is 
necessary  that  many  lines  should  have  their  extremities  at  uniform  distances 
from  the  center  line.  In  laying  off  the  title,  we  determine  what  is  necessary 
to  be  included  in  the  title,  the  space  it  must  occupy,  the  number  of  lines  neces- 
sary, and  the  style  and  arrangement  of  characters  to  be  used.  Thus,  if  the 
title  were,  Plan  of  the  Proposed  Terminus  of  the  Harlem  Railroad  at  New 
York,  1857,  knowing  the  space  to  be  occupied,  we  can  write  the  title  thus  : 


an 


We  now  draw  parallel  lines  at  intervals  suited  to  the  character  of  the  type  we 
intend  to  employ  for  the  different  words.  Harlem  Railroad  is  the  line  to  be 
made  most  prominent ;  this,  calling  the  interval  between  the  words  one  letter, 
includes  15  letters  ;  or,  if  we  consider  /,  with  its  proper  interval,  but  half  a 
letter  (which  will  be  found  a  very  good  rule  in  spacing),  14£  ;  hence  the  center 
of  the  line  will  be  7i  letters  from  the  beginning,  or  \  of  the  space  occupied  by 


TOPOGRAPHICAL  DRAWING.  179 

the  letter  R  and  its  interval.  Draw  a  perpendicular  line  at  the  center,  and 
write  in  R  in  such  a  character  as  may  suit  the  position  to  be  filled,  and  lay  off 
by  letters  and  spaces  the  other  letters.  The  line  Harlem  Railroad  is  intended 
to  occupy  the  whole  length  of  space  ;  that  is,  it  must  be  the  longest  line  in 
the  title,  and  the  lines  above  and  below  must  gradually  diminish,  forming  a 
sort  of  double  pyramid.  Proposed  Terminus  includes  16£  letters  ;  the  /  and 
interval  between  the  words  being  rated  as  above,  we  find  the  center  to  be  nearly 
midway  between  the  words.  These  words,  including  more  letters,  and  being 
confined  within  less  space,  must  be  in  smaller  character  than  the  preceding  ; 
and,  as  a  further  distinction,  a  different  style  should  be  adopted.  Having  de- 
termined this,  we  proceed  to  write  in  the  letters  as  before,  and  in  the  same 
way  with  the  other  lines ;  the  prepositions,  as  unimportant,  are  always  written 
in  small  type. 

i 


.of  the. 


FIOIPOSEB  TE1MIIUS 

of  the 

HARLEM  RAILROAD 


.at. 


NEW  YORK 

1857 


In  general,  it  is  better  that  letters  should  be  first  written  on  a  piece  of  paper, 
distinct  from  the  plan,  as  repeated  trials  may  be  necessary  before  one  is  arranged 
to  suit  the  draughtsman.  Having  formed  a  model  title,  it  may  be  copied  in 
the  plan  by  measures  or  by  tracing  and  transfer  paper.  There  are  some  words, 
such  as  plan,  map,  section,  scale,  elevation,  etc.,  which,  as  they  are  of  constant 
occurrence,  may  be  cut  in  stencil  ;  sometimes  whole  alphabets  are  thus  cut 
and  words  compounded.  It  will  be  found  very  convenient  for  a  draughtsman 
if  he  makes  tracing  or  copies  of  such  titles  as  he  meets  with,  and  preserves 
them  as  models  ;  for  there  is  no  manipulation  on  a  plan  that  contributes  more 
to  the  effect  than  good  lettering  and  arrangement  of  titles,  and  considerable 
practice  should  be  expended  in  acquiring  a  facility  in  lettering,  and,  for  the 
first  start,  perhaps  nothing  will  be  found  more  valuable  than  tracing  good  ex- 
amples. 

We  have  treated  of  mechanical  methods  by  which  most  persons  can  learn 
to  form  letters  and  words  ;  but  it  must  be  borne  in  mind  that  the  distances 
between  letters  on  the  plan  are  only  intended  to  suit  the  eye  ;  if,  therefore,  a 


180 


TOPOGRAPHICAL  DRAWING. 


person  accustom  himself  to  spacing,  so  that  his  eye  is  correct,  there  will  be  no- 
necessity  of  laying  off  by  dividers  ;  in  this  mode,  such  letters  as  A  and  V,  L 
and  T,  are  brought  nearer  each  other  than  the  regular  interval.  In  general,  it 
may  be  observed,  in  reference  to  the  lettering  of  topographical  drawings,  stiff 
letters  like  those  of  stencil  should  not  be  introduced,  but  there  should  be  such 
variety,  incident  on  construction  by  the  pen,  as  may  be  consonant  with  the 
rest  of  the  drawing.  Of  late,  rubber  type  have  been  introduced,  of  fair  forms, 
much  used  on  common  drawings,  by  which  lettering  is  very  rapidly  executed, 
and  is  an  improvement  on  that  of  most  draughtsmen. 


MATEEIALS. 


VAEIED  materials  enter  into  the  composition  of  structures  and  machines, 
or  form  their  supports,  which  are  not  only  to  be  represented  by  the  draughts- 
man, but  he  should  also  understand  the  composition  and  properties  of  these 
materials,  that  he  may  use  them  appropriately  in  his  designs,  and  devise  proper 
forms  to  resist  adequately  and  economically  the  strains  to  which  they  are  to  be 
subjected.  The  earths  and  rocks,  in  their  natural  position,  serve  as  the  sup- 
ports of  structures  and  machines  ;  they  may  be  represented  as  shown  under  the 
head  of  "Topographical  Draw- 
ing," or  by  a  closer  imitation  of 
nature,  with  or  without  color. 

Fig.  306  represents  a  plan  and 
section  of  an  earth-bank  of  a  canal, 
with  a  paved  rock-slope.  A  break- 
water, of  which  the  base  A  is  a 
mass  of  loose  stone,  is  represented 
by  Fig.  307.  A  base  of  rock  may 
be  represented  by  a  stratification 
(Fig.  308).  For  the  foundation 
of  a  structure,  nothing  is  better 
than  solid  rock,  but  the  '  rock 
should  either  have  a  horizontal 
bed  or  be  cut  in  horizontal  steps, 
,so  that  the  walls  resting  on  it 
may  not  slide.  The  base  of  the 
wall  need  not  be  widened.  Sand 
and  gravel  are  also  very  good  foundations,  but  the  base  resting  on  the  earth 
should  in  general  be  about  double  the  width  or  thickness  of  the  wall  rest- 
ing on  it.  For  extensive  buildings  it  is  important  that  the  areas  of  the  bases 


Plan 


FIG.  306. 


J       ........ 

FIG.  307. 


of  its  different  parts  should  be  proportioned  to  the  weights  upon  them,  and 
it  is  also  important  that  soundings  should  be  made  to  determine  whether 


182  MATERIALS. 

there  are  any  compressible  or  sliding  strata  below.  A  stratum  of  3  to  5> 
feet  of  gravel  upon  a  stony  stratum  is  sufficient  foundation  to  support  1  to  1^ 
tons  per  square  foot ;  but,  if  the  sand  rests  upon  rock,  even  at  a  very  great 
depth,  it  is  not  unusual  to  load  it  with  2  to  5  tons  per  square 
foot.  On  sand  and  gravel,  the  building  may  settle  somewhat, 
but  with  proper  bases  uniformly ;  on  wet  clay,  it  is  more  un- 
certain ;  the  building  may  settle  by  displacement,  as  on  a  fluid  ; 
and,  if  the  stratum  is  inclined,  it  is  extremely  apt  to  slide  under 
its  load.  There  are  others  still  more  fluid,  as  quicksand  and 
marsnv  deposits,  where  support  must  be  obtained  by  extending 
the  bases.  On  water  itself,  it  is  obtained  by  means  of  a  scow 
or  tight  box,  the  displacement  being  equal  to  the  weight  of  box  and  structure. 
Earth,  when  first  dug,  occupies  more  space  than  when  in  its  natural  con- 
dition, but,  after  a  time,  it  shrinks  and  becomes  more  compact.  The  earth 
dug  out  of  a  hole,  when  settled,  will  not  fill  the  hole.  Sand,  gravel,  loam,  and 
clay,  will  occupy  from  8  to  12  per  cent  less  space  than  when  in  the  natural  cut. 
Clay  can  be  puddled  to  occupy  25  per  cent  less. 

Loose,  dry  sand  weighs  from  90  to  100  pounds  per  cubic  foot ;  compacted, 
110  ;  gravel,  about  the  same  ;  clay,  120  pounds.  Fresh  water,  at  60°  Fahr.,. 
weighs  about  62 -4  pounds,  and  salt  water  about  64'1  pounds  per  cubic  foot. 
Sands  and  gravels  are  excellent  material  for  embankments  and  fills,  but  clays 
are  much  affected  by  the  weather.  The  slopes  of  the  former  in  cuts  and  fills 
are  usually  1-J  horizontal  to  1  perpendicular  ;  no  fixed  slope  can  be  predicated 
of  clays.  Sands  and  gravels  are  readily  drained,  and,  when  dry,  are  but  little 
affected  by  frost.  The  clays  are  hard  to  drain,  heave  with  the  frost  when  wet, 
and,  under  the  influence  of  a  thaw  or  excess  of  water,  become  fluid.  Very  fine 
sand,  with  gravel,  and  perhaps  some  admixture  of  clay,  forming  the  glacier  till 
of  geologists,  is  known  as  hard-pan\>y  engineers,  very  difficult  to  be  moved  with 
the  pick,  and  often  requiring  blasting.  The  same  material  without  the  gravel 
in  low  bottom  forms  a  quicksand — a  jelly-like  material — from  which,  if  a  spade- 
ful be  taken  out,  the  hole  closes  up  at  once,  and  excavation  shows  but  little 
visible  sign  of  a  depression,  the  space  being  made  good  from  the  entire  mass. 
This  same  material,  dry,  is  a  species  of  hard-pan.  There  is  another  material, 
called  quicksand,  which  is  rather  a  running  sand— even  when  not  wet,  it  rests 
with  a  very  flat  slope  ;  the  particles  are  very  fine,  and  flow  like  the  sands  in  an 
hour-glass. 

Sands  and  gravels  are  large  components  of  mortars,  betons,  and  concrete  ;, 
clay,  of  brick,  tile,  and  pottery. 


BUILDING   MATERIALS. 

The  natural  building  materials  of  civilized  communities  are  wood  and  stone, 
which  are  to  be  worked  or  fashioned  to  the  purposes  to  which  they  are  to  be 
applied. 

Figs.  309,  310,  311,  are  drawings  of  wood,  longitudinal  and  sectional,  in 
which  the  grain  of  the  wood  is  imitated,  but  wood  is  more  often  represented  in 
plain  outline,  and  the  cross-section  of  a  timber  thus  (Fig.  312),  or  by  mere 


MATERI 


183 


FIG.  811. 


FIG.  312. 


hatching.     When  distinguished  by  color,  burnt  sienna  is  used  commonly  for 
wood,  but  sometimes  the  color  of  the  wood  is  imitated. 

The  draughtsman,  for  his  designs, 
will  probably  have  to  confine  himself 
to  the  timber  within  his  reach.  But 
he  should  know  what  is  best  for  his 
purpose,  reference  being  had  to  econ- 
omy in  cost  and  maintenance.  For 
most  purposes,  wood  should  be  sea- 
soned, so  that  joints  may  not  open 
under  this  operation  after  the  ma- 
terial is  in  the  structure.  But,  for 
work  under  water,  wood  should  be 
but  slightly  seasoned,  as  a  swelling 
of  the  wood  may  be  disastrous.  Sea- 
soning of  timber  may  be  done  by  exposure  for  a  time  to  outer  air-currents ;  if  in  a  kiln, 
it  can  be  done  speedily  with  heated  air,  or  by  steam.  For  beams,  girders,  and  the  like, 
there  should  be  few  knots,  especially  on  the  outer  edges— for  posts,  small  ones  are  not 
objectionable;  while  for  sidings  and  under-floors,  firm,  large  knots  do  not  impair  the 
work  ;  but  no  smooth  work  can  be  made  with  knotty  lumber. 

The  trunk  of  the  tree  is  composed  of  sap-wood  and  heart-wood :  the  one  soft,  readily 
rotting;  the  other  more  dense  and  durable.  In  most  specifications,  lumber  is  "to  be 
square-edged,  without  sap,  and  large  or  loose  knots." 

In  selecting  lumber  for  a  permanent  structure,  the  life  and  endurance  of  the  material  are 
to  be  considered.  Most  of  the  woods,  sheltered  from  the  wet  and  exposed  to  air- 
currents,  will  last  for  a  very  long  time ;  but  many  will  check  and  warp  and  become  dis- 
torted. All  lumber  in  earth  beneath  the  level  of  water  will  last  indefinitely.  In  salt 
water,  above  the  earth,  all  are  subject  to  the  attacks  of  the  worm — the  Teredo  and  Lim- 
noria — and,  where  the  water  is  pure,  the  destruction  is  very  rapid.  Sewer-water  and  fresh 
water  are  both  destructive  to  the  worm. 

The  life  of  lumber,  in  situations  exposed  to  wet  and  dry,  can  be  prolonged  by  im- 
pregnating it  with  creosote  or  with  various  metallic  salts,  as  the  chlorides  of  zinc, 
mercury,  pyrolignite  of  iron,  and  others. 


OHAEACTEEISTICS    AND    USE. 

White  Pine. — A  wood  of  the  most  general  application  in  the  market;  is  light,  stiff, 
easily  worked,  nails  are  easily  driven  into  it,  and  takes  paint  well,  warps  and  checks  but 
little  in  seasoning,  endures  well  in  exposed  situations ;  clear  stuff,  of  best  quality,  useful 
for  patterns  and  models,  for  interior  finish  of  houses,  doors,  window-sashes,  furniture.  It 
forms  base  or  inner  core  of  the  best  veneered  work,  holds  glue  well,  and  the  composite 
structure  is  better  than  single  solid  wood.  The  cheaper  kinds  of  pine  are  used  for  frames 
of  buildings,  posts,  girders,  and  beams.  Even  with  large  knots  is  well  adapted  for  board- 
ings, and  is  extensively  used  for  goods-boxes. 

Southern  Pine. — A  heavy,  strong,  resinous,  lasting  wood,  clear  and  mostly  without 
knots,  hard  to  be  worked  by  hand-tools,  and  when  seasoned  difficult  to  nail.  The  surfaces, 
from  their  resinous  character,  do  not  hold  paint  well.  It  is  used  very  largely  for  girders, 
beams,  and  posts  of  mills  and  warehouses,  and  for  floors  of  the  same,  when  exposed  to 
heavy  work  or  travel.  For  the  first,  it  can  be  obtained  of  almost  any  dimension  to  suit; 
for  floors,  it  is  sold  in  long  strips,  from  two  to  six  inches  wide,  of  varied  lengths,  tongued 
and  grooved,  and  when  laid  is  blind-nailed,  toeing  the  nail  through  the  tongue,  so  that  the 
nail-head  does  not  show. 


184:  MATERIALS. 

Canadian  Red,  Norway,  and  Silver  Pines. — Are  resinous  woods,  like  the  Southern 
pine,  and  are  used  for  similar  purposes,  but  are  not  as  valuable — woods  less  straight  in 
the  grain,  and  with  more  knots. 

Spruce. — A  light,  straight-grained  wood,  with  but  few  knots,  which  are  small  and  often 
decayed.  It  does  not  last  well  exposed  to  the  weather,  and  checks  and  warps  badly  in 
seasoning.  It  is  the  most  common  wood  here  for  floor-beams  and  common  floors,  but  it 
must  be  well  braced  and  nailed,  and  is  not  fitted  for  joiner-work. 

Hemlock  is  similar  to  the  spruce,  and,  when  selected,  is  less  liable  to  check  and  twist 
in  seasoning.  It  is  often  of  a  very  poor  quality,  brash  and  shaky.  Exposed,  it  is  but  little 
better,  if  any,  than  the  spruce.  For  stables,  it  is  well  adapted  for  grain-boxes,  as  the  fiber 
prevents  the  gnawing  of  rats. 

Ash. — Some  of  the  ashes  are  of  exceeding  toughness.  A  straight,  close-grained  wood. 
It  is  used  for  carriage  and  machine  frames,  and  for  interiors,  doors,  wainscot,  floors, 
when  no  paint  is  used. 

Chestnut. — Somewhat  like  the  ash  in  appearance,  but  coarser-grained,  and  very  endur- 
ing in  exposed  positions.  It  is  most  largely  used  for  cross-ties  of  railways.  As  a  roof- 
frame  exposed  in  the  inside,  and  in  general  interior  finish  without  paint,  the  effect  is 
very  good.  The  closer-grained  woods  are  very  often  thus  used. 

Black  Walnut. — Is,  in  the  trunk,  a  straight-grained,  gummy  wood,  clogging  the  plane 
a  little  in  its  working ;  the  knots  are  useful  for  veneer.  Were  the  wood  cheap  enough,  it 
would  undoubtedly  make  a  good  frame.  It  is  used  here  for  desks  and  counters,  for  fur- 
niture and  interior  finish,  as  an  ornamental  wood. 

Butternut. — Similar  to  the  black  walnut,  less  commonly  used,  but  fully  equal  as  an 
ornamental  wood. 

Hickory. — A  strong,  tough  wood  ;  is  used  for  cogs  of  mortise-wheels,  handspikes,  axe- 
helves,  and  wheelwrights'  work. 

Beech. — A  close-grained  wood,  but  of  little  application  in  this  market.  Sometimes  used 
for  cogs  of  wheels,  for  small  tool-handles,  and  in  marquetry. 

Oak,  Live. — A  very  strong,  tough,  enduring  wood,  used  industrially  almost  entirely  for 
ship-building.  Ornamentally,  in  marquetry  and  panels. 

Oak,  White.— A  very  valuable,  strong,  tough  wood,  with  great  endurance.  It  is  heavy, 
and  hard  to  work,  and  was  formerly  used  largely  for  the  frames  of  houses,  but  has  been 
superseded  by  the  white  pine.  It  is  u?ed  in  ship-yards  and  in  water-works— for  the  frames 
of  flumes,  penstocks,  and  dams,  and  for  the  planking  of  the  latter,  for  dock-buffers  and 
piles,  and  for  railway  and  warehouse  platforms.  The  red  and  black  oaks  may  in  general 
be  considered  a  cheaper  and  poorer  quality  of  the  white  oak.  All  have  a  handsome  grain, 
that  adapts  them  to  ornamental  work. 

Bans,  Poplar,  White-wood,  are  light  woods,  mostly  used  in  the  manufacture  of  fur- 
niture, for  drawer-bottoms,  cabinet-backs,  panels;  they  are  very  clear  stock,  easily  worked, 
and  can  be  readily  obtained  in  thin,  wide  boards. 

Cedar.— A  straight-grained,  light  wood,  of  great  endurance,  valuable  for  posts,  sills, 
shinies ;  used  for  pails  and  domestic  utensils.  The  red  variety,  from  its  odor,  is  admirable 
for  drawers  and  chests,  preserving  their  contents  from  moths. 

Locust  is  in  the  market  only  in  small  sticks;  is  of  extreme  endurance.  It  is  used 
almost  invariably  here  for  the  sills  of  the  lowest  floors  of  buildings,  where  there  can  be  no 
ventilation,  and  for  treenails  of  ship-planks. 

J57W._  Although  a  tree  of  wide  diffusion,  is  but  little  used  as  lumber.  It  is  kept 
for  an  ornamental  tree,  beyond  its  usefulness  for  any  other  purpose  but  fuel.  Well 
selected,  it  is  said  to  be  an  enduring  timber,  useful  for  piles  and  places  exposed 
to  wet. 

Maples  are  tough,  close-grained  woods,  rather  to  be  considered  among  the  orna- 
mental woods,  for  fnrniture  and  interior  finish.  The  same  may  be  said  of  the  cherry, 


MATERIALS.  185 

plum,  and  apple  tree,  of  which  the  denser  woods  are  admirably  adapted  for  the  handles  of 
small  tools,  for  bushings  of  spools  and  bobbins. 

The  list  of  imported  woods  is  extremely  large,  mostly  for  ornamental  purposes;  but 
the  mahogany  is  one  of  the  very  best  of  woods  for  patterns  and  small  models,  as  it  changes 
but  little  in  seasoning ;  and  the  lignum-vitas,  a  very  hard  and  heavy  wood,  is  used  for  pul- 
ley-sheaves, packing-rings  of  pumps,  water-wheel  steps,  and  shaft-bushings. 

The  weight  and  strength  of  the  several  woods  are  usually  given  in  tables,  but  speci- 
mens of  the  same  wood  differ  essentially  in  both  particulars.  For  all  practical  purposes, 
the  weight  per  cubic  foot  of  white-pine,  spruce,  hemlock,  poplar,  bass,  and  cedar,  may  be 
taken  at  from  23  to  30  pounds.  Ash,  cherry,  chestnut,  black-gum,  black-walnut,  and 
butternut,  from  32  to  45  pounds.  Birch,  beech,  and  elm,  from  40  to  50  pounds.  The 
oaks,  except  live,  from  40  to  55  pounds.  Locust  and  hickory,  from  50  to  60  pounds.  Live- 
oak  and  pitch-pine,  from  60  to  70  pounds.  Lignum- vitse,  from  75  to  80  pounds.  Their 
resistance  to  crushing  varies  from  4,000  to  11,000  pounds  per  square  inch,  und  to  tension, 
from  1,100  to  4,000  pounds,  but  their  practical  use  will  be  given  in  future  illustrations. 

STONES. 

In  selecting  the  form  of  construction,  and  the  stones  of  which  it  is  to  be 
composed,  the  draughtsman  must  be  governed  by  the  fitness  for  the  purpose 
and  the  cost.  He  must  select  from  what  he  can  readily  get,  and  arrange  the 
form  to  suit  the  material.  He  must  know  what  is  to  be  the  exposure,  and 
what  the  effect  will  be  on  the  stones.  Almost  any  stone  will  stand  in  a  pro- 
tected wall,  but  many  of  the  sandstones  and  slates  disintegrate  and  exfoliate 
under  the  influence  of  the  weather,  heat,  cold,  frost,  and  moisture.  Even  the 
granites  are  liable  to  serious  decomposition  when  the  feldspars  are  alkaline  ; 
.and  the  limestones  (dolomites),  of  which  the  English  Houses  of  Parliament  are 
composed,  have  failed  in  the  sulphurous  air  of  London  smoke,  while  at  South- 
well Minster  they  have  stood  for  over  800  years.  Chemical  tests  of  stone  to 
determine  endurance  are  deceptive.  The  safe  way  is  to  see  how  the  material 
has  stood  in  like  situations  to  the  one  in  which  it  is  to  be  employed,  and,  if  this 
is  not  possible,  go  to  the  quarry,  and  see  how  the  stones  have  weathered  there. 

The  strength  of  stones  to  resist  crushing,  as  determined  by  experimental 
cubes,  is  even  in  the  weaker  stones  much  in  excess  of  what  would  be  required 
in  structures,  but  most  stones  are  weak  under  cross-strains,  and  failures  in  con- 
struction are  more  likely  to  occur  by  faulty  workmanship  or  design,  by  which 
the  stones  are  subjected  to  unequal  strains,  and  for  which  they  are  not  adapted. 
The  weight  should  not  be  brought  on  the  outer  edges  or  arrises,  as  the  faces  will 
chip  readily  ;  nor  should  most  stones  be  used  for  wide-span  lintels,  unless  they 
form  a  part  of  the  masonry  above  the  opening,  so  that  the  whole  is  a  beam. 

TECHNICAL   TERMS   OF   MASONRY. 

We  follow  the  nomenclature  recommended  in  "  Transactions  of  the  Ameri- 
can Society  of  Civil  Engineers,"  November,  187?  : 

Rubble  masonry  includes  all  stones  which  are  used  as  they  come  from  the 
quarry,  prepared  at  the  work  by  roughly  knocking  off  their  corners.  It  is 
called  uncoursed  rubble  (Fig.  313)  when  it  is  laid  without  any  attempt  at  regu- 
lar courses  ;  coursed  rubble,  when  leveled  off  at  specified  heights  to  a  horizon- 
tal surface  (Fig.  314). 


186 


MATERIALS. 


FIG.  313. 


FIG.  314. 


Square-stoned  Masonry. — Square  stones  cover  all  stones  that  are  roughly 
squared  and  roughly  dressed  on  bed  and  joints,  and  when  the  joints,  when  laidr 
are  one  half  inch  or  more. 

Quarry-faced  stones  are  those  which  are  left  untouched  as  they  come  from 
the  quarry. 

Pitch-faced  stones  are  those  on  which  the  arris  is  clearly  defined  beyond 
which  the  rock  is  cutting  away  by  pitching-tool. 

Drafted  stones  are  those  in  which  the  face  is  surrounded  by  a  chisel-draft. 


FIG.  315. 


FIG.  316. 


If  laid  in  regular  courses  of  about  the  same  rise  throughout,  it  is  range- 
work  (Fig.  315).  If  laid  in  courses  that  are  not  continuous,  it  is  broken  range 
(Fig.  316). 

Cut  stones  or  ashlar  covers  all  squared  stones  with  smoothly-dressed  bed 
and  joints.  Generally,  all  the  edges  of  cut  stone  are  drafted,  if  the  face  is  not 
entirely  fine  cut,  but  they  may  be  quarry-faced  or  pitch-faced  ;  as  a  rule,  the 


FIG.  317. 


FIG.  318. 


courses  are  continuous  (Figs.  317,  318),  but,  if  broken  by  the  introduction  of 
smaller  stones  of  the  same  kind,  it  is  called  broken  ashlar  (Fig.  316).  If  the 
courses  are  less  than  one  foot  in  height,  it  is  small  ashlar  (Fig.  317). 

Square-stoned  masonry  is  usually  backed  up  with  rubble  masonry.  Any  of 
this  masonry  may  be  laid  dry,  or  with  mortar  or  cement,  which  is  to  be  spe- 
cified. 

The  joints  in  one  course  should  not  come  directly  over  those  of  another  : 


MATERIALS.  1ST 

there  should  be  a  lap  or  bond,  and,  in  connecting  the  front  or  face  with  the 
backing,  headers  must  be  introduced  for  bond.  Headers  are  stones  extending 
into  the  wall,  stretchers  running  with  the  face. 

In  addition  to  the  classes  of  stone- work,  there  is  an  old  form  lately  come 
into  use  called  hock  and  ham  by  old  English  builders  ;  it  is  a  species  of  rub- 
ble, in  which  there  are  no  courses.  The  stones  are  very  carefully  selected  in 
size  and  shape,  so  as  to  make  an  ornamental  work  ;  the  joints  are  close,  but 
have  no  uniformity  of  direction. 

For  rubble-work,  all  varieties  of  sound  stone  are  used,  and  of  almost  any 
size.  In  dry  work,  for  foundations  and  for  heavy  revetment-walls,  the  stones 
are  laid  with  derricks,  but  they  must  have  fair  beds  and  builds.  If  bowlders, 
they  must  be  split,  and  cobbles  in  the  filling  are  worse  than  useless. 

For  rubble  laid  in  mortar,  the  usual  size  is  such  as  can  be  laid  by  hand. 

GRANITIC    STONES. 

Granite  and  syenite  are  by  builders  classed  as  granites.  The  granite  in  general  rifts  in 
any  direction,  and  works  well  under  the  hammer  and  points.  From  these  circumstances- 
it  is  more  desirable  than  the  syenites,  which  are  much  harder  to  be  worked.  Both  are 
admirable  stones  for  heavy  dock-walls,  bridge-abutments,  river-walls,  either  as  rubble- 
squared  stones  or  cut  work,  and  are  very  enduring.  They  are  also  used  for  the  faces  of 
important  buildings,  either  as  fine-cut,  quarry,  or  pitched-face.  Ornamental  work  of  the 
simpler  kind  is  readily  produced  ;  more  elaborate  is  expensive,  but  it  is  about  the  only 
stone  in  this  climate  in  which  foliage  and  sharp  undercut  work  will  stand  the  weather 
without  exfoliating.  These  stones,  especially  the  syenites,  admit  of  a  high  polish,  and  are 
used  considerably  for  columns  and  panels  in  buildings,  and  in  monumental  work.  Gneiss 
is  of  the  granitic  order,  but  a  cheaper,  poorer  stone.  It  splits  with  difficulty,  except  parallel 
with  line  of  bed.  It  has  a  foliated  structure,  and  is  not  adapted  for  ashlar,  but  is  very  good 
for  squared-stone  masonry  and  rubble-work,  and  often  used  for  sidewalk-covers  of  vaults. 

AKGILLACEOUS    STONES. 

The  slates  or  stones  thus  designated  by  builders  were  formerly  in  very  common  use  as 
roofing  material,  and  were  almost  entirely  from  Wales,  but  latterly  they  are  taken  from 
Vermont  and  Pennsylvania,  and  other  parts  of  the  United  States.  They  are  also  used, 
in  thicknesses  of  one  inch  and  above,  for  floors,  platforms,  facing  of  walls,  mantels,  and 
for  wash-tubs  by  plumbers.  Soap-stone  may  be  classed  under  the  clay  stones ;  also,  used 
for  tubs,  for  stoves,  and  for  the  lining  of  grates  and  furnaces. 

The  Ulster,  or  North  River  blue  stone  of  this  market,  is  a  coarser  slate,  a  very  strong 
and  enduring  stone ;  it  can  be  quarried  of  varying  thickness  up  to  twelve  inches,  and  of 
any  dimension  that  can  be  transported.  It  can  be  readily  cut,  hammer-dressed,  axed, 
planed,  and  rubbed.  Is  generally  used  for  sidewalks  under  these  various  forms.  It  ia 
used  as  bond-stones  in  brick  piers,  for  caps,  sills,  and  string-courses. 

THE   SANDSTONES. 

Sandstones,  called  also  freestones,  from  the  ease  with  which  they  are  worked ;  and  from 
their  colors,  are  very  popular  for  the  fronts  of  edifices.  In  general,  they  are  not  very 
enduring  stones,  and  when  laid  must  be  set  parallel  to  their  natural  beds,  as  otherwise 
they  flake  off  under  the  influence  of  the  weather.  The  sandstones  are  not  all  of  the 
same  quality;  those  in  which  the  cementing  material  is  nearly  pure  silex,  are  strong, 
enduring  stones,  but  not  those  in  which  the  cementing  material  is  alumina,  or  lime.  By 
examining  a  fresh  fracture,  the  character  of  the  stone  can  generally  be  detected.  A  clear, 
shining  surface  with  sharp  grains  indicate  a  good  stone;  while  rounded  grains,  a  dull. 


188  MATERIALS. 

mealy  surface,  indicate  a  soft,  perishable  stone.  None  of  the  sandstones  in  this  locality 
are  used  for  heavy  pier  or  abutment  work  and  the  like,  but  there  are  sandstones  in  other 
localities  adapted  to  it. 

LIMESTONE. 

The  coarser  calcareous  stones  are  of  great  variety ;  some  are  well  adapted  for  building 
stones,  being  hard  and  compact,  while  others  are  soft  and  friable.  They  are  more  easily 
worked  than  granite,  but  are  not  considered  as  enduring.  They  are  well  adapted  to  the 
same  class  of  heavy  work,  and  the  locks  of  the  Erie  and  Northern  Canals  and  the  dam 
across  the  Mohawk,  at  Cohoes,  are  built  from  limestone  on  the  line  of  the  canals. 

The  finer  kinds  of  limestones  are  classed  under  the  head  of  marbles.  They  are  easily 
worked,  sawed,  turned,  rubbed,  and  polished.  Marble  is  not  popular  as  a  building  material, 
although  more  enduring  than  most  sandstones,  but  is  susceptible  to  the  action  of  sulphurous 
gases  in  the  smoky  air  of  cities ;  and  it  is  said  that  the  Capitol  at  Washington,  D.  C., 
built  of  marble,  is  suffering  from  disintegration.  But,  for  interior  finish,  as  tiles,  wain- 
scots, architraves,  mantels,  linings  of  walls,  it  is  admirably  adapted,  and  from  its  richness, 
deanliness,  and  variety  of  color,  it  is  very  ornamental  and  effective. 

ARTIFICIAL  BUILDING   MATEEIAL. 

The  most  common  and  useful  are  bricks.  They  are  generally  made  of  clay,  with  an 
admixture  of  sand,  well  incorporated  together,  and  mixed  with  water  to  the  consist- 
ency of  a  smooth,  strong,  viscous  mud,  pressed  into  molds,  dried,  and  burned,  *the  best 
quality  being  those  in  the  interior  of  the  kiln.  The  exteriors  are  light,  friable  bricks 
adapted  to  walls  supporting  but  little  weight  and  not  exposed  to  wet.  The  brick  forming 
the  arches  are  very  hard-burned,  dark  in  color,  often  swelled  and  cracked ;  but  by  proper 
selection,  they  can  be  used  for  foot-walks.  A  good  brick  is  well  burned  throughout ; 
when  struck,  it  gives  a  ringing  sound,  and  is  of  uniform  shape. 

Bricks  vary  somewhat  in  size  and  weight  in  different  localities — from  8  to  8^  inches 
long  x  3£  to  4  inches  broad  x  2  to  2£  inches  thick ;  in  general,  the  thickness  of  a  wall 
with  the  joints  is  called  some  multiple  of  4",  as  8",  12",  16".  Here  an  8-inch  wall  by  1 
foot  face  contains  14  bricks ;  12-inch,  21  ;  16-inch,  28  bricks — 9  courses  high  are  equal  to  2 
feet.  In  the  Eastern  States  the  brick  is  somewhat  thinner — 5  courses  to  the  foot.  The 
best  front  brick  are  pressed,  and  are  a  little  larger  than  the  common  brick.  Philadelphia 
and  Baltimore  pressed  brick  are  distinguished  by  their  clear,  cherry-red  color ;  Milwaukee 
are  of  a  pale-straw  color. 

Bricks  are  laid  in  mortar,  of  lime,  lime  and  cement,  or  cement  only— all  with  an  ad- 
mixture of  sand  ;  in  common  walls,  in  lime  ;  in  walls  of  heavy  buildings,  above-ground,  in 
lime  and  cement ;  beneath,  and  in  wet,  exposed  positions,  in  cement  only.  The  common 
bond  of  the  different  courses  of  brick  is  by  header-courses  every  fifth  or  seventh  course. 
When  bricks  are  laid  in  arches  they  are  set  on  edge,  and  turned  in  4-inch  rings,  sometimes 
without  any  bond  between  the  different  rings;  sometimes  with  a  bond  of  brick  length- 
ways, when  two  courses  come  on  the  same  line. 

Bricks  set  on  edge,  as  in  arches  or  in  a  level  course,  are  here  termed  rollocks. 

Arch  brick,  between  iron  beams,  to  reduce  the  weight,  are  often  made  hollow,  and  laid 
in  flat  arches ;  that  is,  the  joints  are  radial,  but  the  upper  and  lower  surfaces  are  level. 
Hollow  brick  are  also  used  for  walls  and  partitions. 

Fire-brick  can  be  made  of  any  size  and  pattern,  but  are  usually  9  x  4£  x  2f.  They  are 
used  for  the  lining  of  furnaces,  flues,  and  chimneys,  exposed  to  the  action  of  flame  or  great 
heat.  Fire-clay,  with  an  admixture  of  sawdust,  which  is  burned  out  in  the  firing,  leaves  a 
light,  porous,  spongy  mass,  which  can  be  sawed  in  sheets  or  strips,  and  is  well  adapted 
for  covering  the  exposed  parts  of  iron  beams  and  girders,  and,  as  it  admits  of  nailing,  is 
convenient  for  partitions. 

Enameled  Brick. — The  English  size  is  that  of  fire-brick — the  American  is  that  of  com- 


MATERIALS.  189 

mon  brick.  The  brick,  on  the  faces  to  be  exposed,  are  covered  with  glaze  of  varied  colors 
and  designs,  and  fired.  They  make  a  handsome  ornamental  face  for  walls,  do  not  absorb 
moisture,  and  can  be  washed. 

Tile  are  a  species  of  brick,  with  or  without  enamel.  The  latter  were  originally  used 
for  roof-covering,  but  now  are  used  in  flooring  walks  and  the  like.  The  enameled  or 
encaustic  tile  are  generally  in  squares,  4"  x  4",  6"  x  6",  8"  x  8",  but  there  are  smaller 
ones  for  tessellation,  and  rectangular  strips  for  borders.  They  can  be  obtained  of  any 
color  or  design,  forming  beautifully  ornamented  floors  and  wall-panels. 

Terra-cotta,  a  kind  of  brick,  is  now  largely  used  for  exterior  decoration.  It  is  molded 
in  every  variety  of  capitals,  cornices,  caps,  friezes,  and  panels.  It  is  a  good,  strong  brick, 
with  all  the  good  qualities  of  such  a  material. 

Mortars. — Brick  are  never  laid  dry,  except  in  the  under  part  of  drains,  to  admit  of  the 
removal  of  ground-water.  Stone-work,  except  in  rough,  heavy,  rubble-work,  is  also  gen- 
erally  laid  in  mortar.  Where  cut-work  is  backed  with  rubble,  the  joints  in  the  latter 
should  be  as  close  as  possible,  and  full  of  mortar,  that  the  settling  of  the  wall  in  itself 
may  not  be  more  in  the  backing  than  in  the  face.  Some  lay  the  rubble  dry,  and  fill  in 
with  cement  grout,  or  cement  mortar  made  liquid  to  flow  into  the  interstices,  but  the  sand 
is  apt  to  separate  and  get  to  the  bottom  of  the  course. 

By  mortar,  is  usually  understood  a  mixture  of  quicklime  and  sand,  but  mortar  may 
have  an  addition  of  cement  to  the  lime,  or  it  may  be  cement  only  with  sand. 

Lime,  or  properly  quicklime,  is  made  by  the  calcination  of  limestone,  shells,  and  sub- 
stances composed  largely  of  carbonate  of  lime,  carbonic-acid  gas,  water  of  crystallization, 
and  organic  coloring-matter.  Quicklime,  brought  in  contact  with  water,  rapidly  absorbs  it, 
with  a  great  elevation  of  temperature,  and  bursting  of  the  lime  into  pieces,  reducing  it  to 
a  fine  powder,  of  from  two  to  three  and  a  half  times  the  volume  of  the  original  lime.  This 
is  slaked  lime.  It  may  be  slaked  slowly  by  exposure  to  the  air,  from  which  it  will  take 
the  moisture.  This  is  air-slaked  lime.  Barrels  of  lime  exposed  to  rain  often  take  fire 
from  the  heat  caused  by  slaking.  The  paste  of  slaked  lime  may  be  kept  uninjured  for  a 
considerable  time,  if  protected  from  the  air,  and  this  may  readily  be  done  by  a  covering  of 
sand,  and  it  is  customary,  in  some  places,  to  hold  it  over  one  season,  as  an  improvement 
to  the  uniformity  of  quality  in  the  paste.  But,  in  general,  the  lime  is  used  soon  after 
slaking,  and  is  thoroughly  mixed  with  sand,  in  various  proportions,  generally  about  two 
of  sand  to  one  of  lime.  The  theory  of  the  mixture  is,  that  the  lime  should  fill  the  void 
spaces  in  the  sand,  and  the  space  occupied  by  the  mortar  is  a  little  in  excess  of  that  occu- 
pied by  the  sand  alone. 

The  sand  should  be  sharp,  clean,  silicious  grains,  from  one  twelfth  to  one  sixtieth  of  an 
inch  in  diameter.  Close  brick-joints  do  not  admit  of  as  coarse  sand  as  those  of  cut  stone 
work,  and,  in  rubble-work,  sand  coarser  than  the  above  can  be  used,  and  there  will  be 
considerable  saving  of  lime  in  using  a  mixture  of  coarse  and  fine  sand. 

The  hydraulic  limes  contain  a  small  proportion  of  silica,  alumina,  and  magnesia  ;  slake 
with  but  little  heat,  and  small  increase  of  volume  ;  are  more  or  less  valuable,  according" 
to  the  property  which  they  have  for  hardening  under  water ;  but,  in  this  particular,  are 
not  equal  to  the  hydraulic  cements,  which  contain  a  larger  proportion  of  silica,  alumina, 
and  magnesia.  They  are  made  by  calcining  natural  rocks,  or  Jby  the  combination  of  clay 
and  soft  carbonate  of  lime,  or  chalk,  calcining  and  grinding.  The  cements  make  a  paste 
with  water,  with  little  or  no  heat  on  slaking,  and  set,  in  open  air  or  under  water,  with 
more  or  less  rapidity  ;  but  this  is  not  a  sure  criterion  of  the  value  of  the  cement,  when  time 
comes  in  as  an  element  before  the  work  is  subject  to  stress. 

Cement  is  mixed  with  sand  in  varied  proportions — from  1  to  1  to  1  to  3 — it  is  stronger 
without  any  admixture  of  sand,  but  is  seldom  used  neat,  except  in  pointing,  and  for  very 
close  joints.  By  experiments  of  Mr.  F.  0.  Norton  (Trans.  Am.  Soc.  of  C.  E's.)  it  was 
found  that  Portland  cement,  with  two  volumes  of  sand,  was  equal  to  that  of  Rosendale  (or 


190  MATERIALS. 

native  cement)  with  one  part  of  sand.  In  purchasing  cement,  it  is  usual  that  it  should  be 
required  to  be  up  to  a  certain  standard ;  that  is,  made  np  into  a  ball  with  water  at  65° 
temperature;  it  should  set  in  water  to  withstand  a  pressure  of  say  one  pound  on  a  one- 
quarter  inch  wire  within  so  many  minutes.  By  many,  cements  are  required  to  be  of  a 
oertain  degree  of  fineness — that  only  a  very  small  per  cent  should  be  left  on  the  screen,  say 
of  one  sixtieth-inch  mesh,  and  that  it  should  weigh  about  80  pounds  to  the  bushel,  and 
that  it  should  have  a  certain  tensile  strength  after  so  long  a  set. 

Cement  is  used  in  all  masonry  in  exposed  and  wet  situations.  With  a  small  admixture 
of  lime,  it  works  better  under  the  trowel,  and  for  brick-work  it  does  not  sensibly  impair 
its  value.  Cement  adds  to  the  strength  of  lime-mortar,  and  gives  it  an  amount  of  hydrau- 
licity.  To  increase  the  quick-setting  of  cement,  it  may  sometimes  be  necessary  to  add  a 
little  plaster-of-Paris,  but  it  is  preferable  to  get  a  quick-setting  cement. 

Concrete  or  beton  are  terms  now  used  for  the  same  material.  It  consists  of  cement, 
sand,  and  gravel,  or  broken  stone,  which  may  be  intimately  mixed,  in  varied  proportions, 
according  to  the  quality  of  the  cement  and  the  character  of  the  inert  materials.  For  the 
blocks  of  the  New  York  city  docks,  the  proportions  were  : 

Portland  cement 3  volumes. 

Sand,  damp 5 

Broken  stone 10         " 

This  is  a  strong  mixture.  It  is  not  uncommon  to  make  a  cement  of  Rosendale  cement  1, 
sand  2  to  3,  and  broken  stone  or  clean  gravel  as  much  as  can  be  well  covered  by  the  mix- 
ture, but  it  should  have  time  to  set. 

Concrete  is  used  for  the  base  course  or  foundations  of  walls,  and  is  formed  in  situ, 
that  is,  depositing  and  ramming  it  in  the  trench  where  it  is  to  be  left;  or  by  forming  in 
molds,  in  immense  blocks,  for  docks  or  break-water,  or  in  the  smallest  forms  of  brick  and 
moldings. 

The  bituminous  cements  are  formed  of  natural  bitumens,  or  artificial  from  coal-tar 
mixed  with  various  proportions  of  gravel  and  inert  material.  The  mixture  is  usually 
heated,  put  down  in  layers,  and  rolled  or  rammed.  It  is  used  for  roads  and  sidewalks, 
and  for  water-proof  covering  of  vaults.  For  the  covering  of  roofs,  coarse  paper,  sat- 
urated with  bitumen,  is  put  on  in  layers,  one  over  the  other,  breaking  joints,  cemented 
with  the  bitumen,  the  last  coat  being  of  bitumen,  in  which  gravel  is  imbedded.  For 
an  anti-damp  course  in  a  wall,  or  for  the  joints  in  the  bricks  of  a  wet  cellar-floor, 
or  on  top  of  a  roof,  bitumen  is  used  as  a  cementing  material — the  bricks  must  be  dry, 
bitumen  hot. 

Plastering. — Coarse-stuff  is  nothing  more  than  common  brick-mortar,  with  an  admix- 
ture of  bullock's  hair.  When  time  can  not  be  given  for  the  setting  it  is  gauged,  that  is, 
mixed  with  some  plaster-of-P;iris.  Fine-stuff  is  made  of  pure  lump-lime  with  an  admix- 
ture of  fine  sand,  and  perhaps  plaster-of-Paris.  Hard-finish  is  composed  of  fine-stuff  and 
plaster-of-Paris.  One-coat  work  is  of  coarse-stuff,  which  may  be  rendered,  that  is,  put  on 
masonry,  or  laid  on  laths.  Two-coat  work  is  a  coat  of  coarse-stuff,  or  scratch,-coat ;  that 
is,  after  the  coat,  is  partially  dry  it  is  scored  or  scratched  for  a  back  for  the  next  or  fine 
coat.  In  three  coats,  the  first  coat  is  a  scratch-coat,  the  second  the  brown-coat,  and  the 
third  is  hard-finish,  or  stucco.  Keene^s  cement,  for  the  last  finish,  gives  a  very  hard 
surface,  which  admits  of  washing. 

A  single  brick  weighs  between  4  and  5  pounds ;  but  a  cubic  foot,  well  laid  in  cement, 
with  full  joints,  will  weigh  about  112  pounds.  They  have  resisted,  in  an  experimental 
test,  as  high  as  13,000  pounds  to  the  square  inch,  but  12  tons  should  be  the  limit  to  the 
load  per  square  foot ;  and  the  brick  should  be  uniform,  well  burned,  and  closely  laid  in 
cement,  and  without  cross-strain.  In  lime  mortar,  the  load  should  not  exceed  3  tons  per 
square  foot. 


MATERIALS. 


191 


The  granites  weigh  from  160  to  180  pounds  per  cubic  foot ;  the  limestones  from  150 
to  175  ;  the  sandstones  from  130  to  170  ;  the  slates  from  160  to  180 ;  mortar,  set,  about 
100  pounds ;  masonry,  laid  full  in  mortar,  according  to  the  quality  of  the  stone  and  the 
percentage  of  mortar,  from  150  to  170  pounds.  Some  of  the  granites  have  withstood  a 
crushing  strain  of  15,000  pounds  per  square  inch,  and,  when  structures  are  important,  and 
subject  to  great  strains,  specimens  of  the  stones  to  be  employed  should  be  tested ;  but,  for 
practical  purposes,  common  mortar-rubble  is  not  considered  equal  in  strength  to  a  brick 
wall,  as  it  is  seldom  laid  with  equal  care,  and  the  joints  are  not  as  likely  to  be  well  filled, 
and  the  load  as  evenly  distributed ;  but  cut  stones  will  sustain  more,  and  ashlar,  up  to  50 
tons  per  square  foot  for  sound,  strong  stones. 

METALS. 

Metals  are  often  to  be  shown  distinctively  by  the  draughtsman.  If  lie  can 
use  color,  he  will  in  a  measure  imitate  that  of  the  material.  For  cast-iron, 
India-ink,  with  indigo,  and  a  slight  admixture  of  lake  ;  for  wrought-iron,  the 
same  colors,  with  stronger  predominance  of  the  blue  ;  steel,  in  Prussian-blue  ; 
brass,  in  a  mixture  of  gamboge  and  burnt  sienna  ;  copper,  gamboge  and  crim- 
son lake.  But  it  is  often  requisite  to  express  distinctive  metals  in  drawings 
where  no  color  is  admissible.  When  the  drawings  may  be  required  for  photo- 
graphing, or  reproduced  in  printing,  some  conventional  hatchings  are  used  to 
represent  sections  of  metals,  but  none  have  been  so  established  as  to  have  a 
universal  application.  The  following  are  submitted  to  represent  the  most  com- 
mon industrial  metals  : 


^\VS\*N$ 


Cast  Iron. 
FIG.  319. 


Wrought  Iron. 
FIG.  320. 


Steel. 
FIG.  321. 


Brass. 
FIG.  322. 


Lead. 
FIG.  323. 


Under  the  term  iron  may  be  included  cast-iron,  wrought-iron,  and  steel,  differing  from 
"each  other  in  the  percentage  of  carbon  contained,  and  in  the  uses  to  which  they  are  applied. 
•Oast-iron  contains  more  carbon  than  the  others,  say  from  two  to  five  per  cent.  It  can 
be  cast  in  varied  forms  in  molds,  but  can  not  be  welded  or  tempered.  The  usual  molds  are 


192 


MATERIALS. 


in  sand  or  loam,  in  which  the  pattern  is  imbedded,  and  when  drawn  out  the  space  is  filled 
with  molten  metal.  The  drawing  of  patterns  for  molding  involves  a  knowledge  of  the  art 
of  founding.  The  shrinkage  of  the  metal,  usually  about  one  per  cent,  for  which  provision 
must  be  made  in  increased  size  of  pattern,  is  provided  for  by  the  pattern-maker,  the 
draughtsman  giving  finished  sizes,  but  the  draughtsman  must  know  whether  the  pattern 
can  be  drawn  from  the  sand,  and  by  what  system  of  cores  voids  can  be  left ;  or  it  may 
often  happen  that  castings,  designed  as  a  whole,  will  have  to  be  made  in  a  number  of 
pieces,  involving  flanges  and  bolts.  In  cooling,  the  shrinkage  takes  place  the  soonest  in 
the  thinnest  parts,  and,  if  great  care  be  not  taken  by  the  molder  in  exposing  the  thicker 
parts  to  the  air  first,  the  parts  will  shrink  unequally,  and  there  will  be  a  strain  induced 
which  will  materially  weaken  the  casting,  and  it  may  even  break  in  the  mold.  The 
draughtsman,  in  his  design,  should  make  the  parts  of  as  uniform  thickness  as  possible. 

Castings  cool  from  the  outside  inward,  in  annular  crystals  perpendicular  to  the  face, 
as  in  Figs.  324  and  325.  Now,  if  the  casting  consist  of  a  right  angle  (Fig.  326),  there  will 
evidently  be  a  weak  place  along  the  line  A  B,  but,  if  the  angle  be  eased  by  a  curve,  the 


FIG.  324. 


FIG.  325. 


FIG.  326. 


FIG.  327. 


crystallization  takes  place  as  in  Fig.  327,  and  the  line  of  weakness  is  avoided.  This  is 
effected  by  a  very  small  easement  of  the  angle,  and  a  cove  is  almost  invariably  introduced. 
In  castings,  in  almost  all  metals,  the  same  effects  result  from  cooling,  and  therefore  the 
changes  of  direction  should  not  be  abrupt. 

"When  castings  are  ordered  for  important  structures,  iron  of  certain  tensile  strength  is 
called  for,  and  specimens  of  the  metal,  in  small  rectangular  bars,  are  required,  cast  at  the 
same  time  and  under  as  nearly  the  same  conditions  as  the  casting  which  may  be  subjected 
to  test. 

If  the  casting  be  made  in  dry  sand,  it  cools  slowly,  and  the  surface  is  comparatively 
soft;  if  in  greensand — sand  somewhat  moist — the  surface  becomes  harder;  but  if  cast 
on  an  iron  plate,  or  chill,  some  irons  become  as  hard  as  the  hardest  steel,  useful  in 
surfaces  exposed  to  heavy  wear,  as  the  treads  of  rail  way- wheels.  Cast-iron,  in  general, 
is  brittle  under  the  blows  of  a  hammer,  but  some  mixtures,  under  a  process  of  annealing, 
become  malledble  iron,  used  largely  for  steam -fittings,  parts  of  agricultural  machines,  forms 
requiring  the  toughness  of  wrought-iron,  but  difficult  to  forge. 

\\  rought-iron  is  produced  from  cast-iron  by  removing  the  carbon  and  impurities  by 
puddling,  squeezing,  heating,  and  rolling.  As  a  material,  it  is  sold  in  all  sizes  of  wire, 
rods,  shafts,  bars,  plates,  shapes— girders  and  beams,  chains  and  anchors.  Its  applica- 
tion industrially  is  well  known.  When  hot,  it  can  be  welded,  forged,  drawn,  and  swaged 
into  almost  any  required  shape.  Under  the  steam-hammer,  the  largest  shafts,  anchors,  -and 
cranks  can  be  built,  or  by  hand  or  by  machinery  it  can  be  wrought  into  tacks,  nuts,  bolts,, 
nails,  or  drawn  into  the  finest  wire. 

For  shafts  of  mills  it  is  generally  turned  in  a  lathe  and  polished,  but  of  late  it  can  he- 
bought,  up  to  four  inches  diameter,  cold-rolled,  which  adds  very  considerably  to  the  strength, 
and  is  ready  for  use. 

Bessemer  and  Siemens-Martin  metals  are  made  by  burning  out  the  carbon  from  a 
melted  iron,  and  then  reintroducing  a  known  quantity,  say  from  0'03  to  0'6  per  cent  of  car- 
bon. There  are  other  patents  covering  somewhat  different  irons,  but  the  above  are  the  best 
known.  All  are  commonly  classed  as  steel,  but  by  many  are  called  homogeneous  metal : 


MATERIALS.  193 

first-class  iron,  of  very  uniform  texture  and  great  strength,  but  not  equal  to  that  of  the  best 
steel. 

Steel  is  produced  from  pure  wrought-iron  by  what  is  called  cementation — heating  the 
bars  in  contact  with  charcoal,  by  which  a  certain  amount  of  carbon  is  taken  up.  The  bars, 
when  taken  out,  are  covered  with  blisters,  apparently  from  the  expansion  of  minute  bub- 
bles within  ;  hence  called  blistered  steel.  From  this  shear-steel  can  be  produced  by  piling, 
heating,  and  hammering,  or  cast-steel  from  melting  in  a  crucible. 

Steel,  when  broken,  does  not  show  the  fibrous  character  of  wrought-iron.  The  frac- 
ture of  shear-steel  is  fine,  with  a  crystalline  appearance.  The  fracture  of  cast-steel  is  very 
fine,  requiring  very  close  inspection  to  show  the  crystals  or  granulations ;  its  appearance 
is  that  of  a  fine,  light,  slaty -gray  tint,  almost  without  luster.  Steel  is  stronger  than  any  of 
the  other  iron  products,  and  especially  applicable  for  the  piston-rods  of  steam-engines,  and 
positions  requiring  great  strength  and  stiffness,  with  the  minimum  of  space.  But  it  is  the 
way  in  which  steel  can  be  hardened  and  tempered  which  adapts  it  to  its  peculiar  appli- 
cations. 

When  the  malleable  metals  are  hammered  or  rolled,  they  generally  increase  in  hard- 
ness, elasticity,  and  denseness,  and  some  kinds  of  steel  springs  are  made  by  the  process 
of  hammer-hardening ;  but  the  usual  process  of  hardening  and  tempering  is  by  heating 
the  steel  to  a  degree  required  by  the  use  to  which  it  is  to  be  applied,  and  cooling  it 
more  or  less  suddenly  by  immersing  in  water  or  oil.  The  greater  the  difference  between 
the  heated  steel  and  the  cooling  medium,  the  greater  the  hardness,  but  too  much  heat 
may  burn  the  steel,  and  too  sudden  cooling  make  it  too  brittle.  Steel,  in  tempering,  is 
heated  from  430°  Fahr.  to  630°.  The  temperature  is  shown  by  the  color — from  a  pale 
yellow  to  deeper  yellow,  light  purple  to  a  dark  purple,  dark  blue  to  a  light  blue,  with  a 
greenish  tinge. 

Steel  is  used  for  the  edges  of  all  cutting-tools,  faces  of  hammers  and  anvils,  and  is  gen- 
erally welded  to  bodies  of  wrought-iron,  but  often  composing  the  entire  tool ;  for  saws, 
springs,  railway  tires,  pins,  and  can  be  bought  in  the  form  of  wire,  rods,  bars,  sheets,  and 
plates,  in  varied  forgings  and  castings. 

All  irons  are  very  liable  to  rust,  and  must  be  protected  where  exposed  to  moisture. 
Polished  surfaces  are  kept  wiped  and  oiled,  others  painted,  others  galvanized  or  plated 
with  some  less  oxidizable  metal,  generally  tin,  zinc,  or  nickel.  Of  late,  a  process  has 
been  introduced  of  coating  them  with  black  oxide,  but  is  yet  of  no  general  application. 

Antimony,  bismuth,  copper,  lead,  tin,  and  zinc,  are  used  more  or  less  industrially,  and 
alloys  of  them  are  extremely  useful.  They  may  be  hardened  somewhat  by  the  process 
of  rolling  and  hammering,  but  can  not  be  welded.  Joinings  are  made  by  soldering  or 
brazing  or  burning — that  is,  melting  together. 

Antimony  expands  by  cooling.  With  tin,  in  equal  proportions,  it  makes  speculum- 
metal,  and  is  used,  with  lead,  to  make  type.  Type  metal  makes  a  very  good  bearing  for 
shafts  and  axles. 

Bismuth  is  chiefly  used  as  a  constituent  of  fusible  metal :  3  bismuth,  5  lead,  and  3  tin, 
is  an  alloy  which  melts  at  212°.  Other  mixtures  are  made,  increasing  the  melting-point 
to  adapt  the  metal  for  fusible  plugs  in  boilers,  or  lowering  the  melting-point,  so  that,  in 
case  of  fire  in  a  building,  a  heat  of  say  140°  melts  the  joint  made  by  the  metal,  and  lets 
water  through  sprinklers,  to  automatically  put  out  the  fire. 

Copper  is  very  malleable  and  ductile.  In  sheets,  it  is  used  for  the  cover  of  roofs,  gut- 
ters, leaders,  lining  of  bath-tubs,  kettles,  stills,  and  kitchen  utensils.  It  is  worked  more 
easily  than  iron,  and  is  stronger  than  lead  or  zinc,  but  it  is  much  more  costly  than  either 
of  these  metals,  and  its  oxide  is  so  poisonous  that,  without  great  care  and  cleaning,  it  can 
not  be  used  to  transmit  or  contain  anything  that  may  be  used  as  food,  without  a  cover  of 
tin.  It  oxidizes  slowly,  and  is  used  extensively  for  ships'  fastenings  and  for  bottom-sheath- 
ing. It  is  the  most  important  element  in  all  the  brass  and  bronze  alloys. 
13 


194  MATERIALS. 

Brass,  in  common  use,  covers  most  of  the  copper  alloys,  no  matter  what  the  other 
components  are,  whether  zinc,  tin,  or  lead,  or  all  three. 

Copper  and  zinc  will  mix  in  almost  any  proportions.  The  ordinary  range  of  good 
yellow  brass  is  from  4|  to  9  ounces  of  zinc  to  the  pound  of  copper.  With  more  zinc  it 
becomes  more  crystalline  in  its  structure,  but,  as  zinc  is  very  much  cheaper  than  copper, 
the  founder  is  apt  to  increase  the  percentage  of  zinc,  with  the  addition  of  a  small  per- 
centage of  lead.  Muntz  metal,  in  its  best  proportion,  contains  lOf  ounces  of  zinc  to  the 
pound  of  copper. 

Copper  and  tin  mix  in  almost  any  proportion.  The  composition  of  ancient  bronzes  is 
from  1  to  3  ounces  of  tin  to  the  pound  of  copper.  Ten  parts  of  tin  to  90  of  copper  is  the 
usual  mixture  for  field-pieces,  and  this  is  used  in  steam-engine  work,  often  under  the 
name  of  composition.  Bell-metal  is  from  4  to  5  ounces  of  tin  to  the  pound  of  copper ;  Bab- 
bit-metal, for  journal-boxes,  90  of  tin  to  10  of  copper. 

Copper  and  lead  mix  in  any  proportion  up  to  nearly  one  half  lead,  when  they  separate 
in  cooling. 

An  addition  of  from  one  quarter  to  one  half  ounce  of  tin  to  the  pound  of  yellow  brass 
renders  it  sensibly  harder.  A  quarter  to  one  half  ounce  of  lead  makes  it  more  malleable. 

German-silver  is  50  copper,  25  zinc,  and  25  nickel. 

Holzapfel  gives  the  following  alloys  : 

1-J-  ounce  tin,  £  ounce  zinc,  to  16  ounces  copper,  for  works  requiring  great  tenacity. 

1-J-  to  If  ounces  tin,  2  ounces  brass,  to  16  ounces  copper,  for  cut  wheels. 

2  ounces  tin,  1-|  ounce  brass,  to  16  ounces  copper,  for  turning-work. 

2J  ounces  tin,  1£  ounce  brass,  to  16  ounces  copper,  for  coarse-threaded  nuts  and  bearings. 

2£  ounces  tin,  2£  ounces  zinc,  to  16  ounces  copper,  Sir  F.  Chantry's  mixture,  from  which 
a  razor  was  made,  nearly  as  hard  as  tempered  steel. 

Professor  R.  H.  Thurston,  of  Stevens  Technological  Institute,  has  tested  various  alloys 
of  copper,  tin,  and  zinc,  and,  by  a  graphic  method,  determines  the  best  alloy  for  toughness 
as  well  as  strength  to  be — copper  55,  tin  2*5,  zinc  44'5. 

There  are  various  other  alloys,  as  phosphate  bronze,  aluminium  bronze,  Sterro-metal,  of 
which  the  strength  will  be  given  hereafter  in  a  table. 

Lead  is  a  very  soft  metal,  that  can  be  readily  rolled  into  sheets  and  drawn  into  pipes, 
and  is  so  flexible  that  it  can  be  readily  fitted  in  almost  any  position.  It  is,  therefore, 
especially  adapted  to  the  use  of  plumbers,  for  the  lining  of  cisterns  and  tanks,  and  for  pipes 
for  the  conveyance  X)f  water  and  waste.  For  pipes  for  conveying  pure  water  for  drinking 
purposes,  or  for  cisterns  containing  it,  it  is  objectionable,  as  it  oxidizes,  and  the  oxide  is  a 
dangerous  and  a  cumulative  poison,  but,  in  common  waters  which  are  more  or  less  hard, 
the  insides  of  the  pipes  become  covered  with  a  deposit  which  protects  them.  It  is  well, 
before  drinking  from  a  lead  pipe  in  which  the  water  has  stood  for  a  time,  to  draw  off  all 
the  water,  and,  in  lead-lined  cisterns  exposed  more  or  less  to  the  air,  to  protect  them  by  a 
coating  of  asphalt  varnish.  Lead  expands  readily,  and  has  so  little  tenacity  that,  in  many 
positions,  if  heated,  it  has  not  strength  in  cooling  to  bring  it  back  to  its  original  position. 
It  remains  in  wrinkles  on  roofs,  and,  for  pipes  conveying  hot  water,  unless  continuously 
supported,  it  will  hang  down  in  loops,  continuously  increasing  under  variations  of  tem- 
perature, to  rupture.  But  it  makes  a  very  good  plating  for  sheet-iron  for  roofs,  and  its 
oxides  are  the  most  valuable  of  all  pigments. 

Tin,  in  a  pure  state,  is  used  for  domestic  utensils,  as  block-tin,  and  has  also  been  used 
for  pipes  in  the  conveyance  of  water  by  parties  who  feared  the  poisonous  qualities  of  lead 
pipe.  But  its  chief  use  is  for  the  covering  of  sheet-iron,  which  is  sold  under  the  name  of 
tin  or  tin-plate,  and  is  of  universal  application  for  architectural,  industrial,  and  domestic 
purposes.  Its  oxide  is  not  injurious,  and  it  is  so  little  affected  by  air  and  moisture  that 
roofs,  in  many  places,  covered  with  it,  need  no  painting,  and  oxidization  takes  place  in  the 
iron  beneath  only  from  deficiency  in  plating,  or  from  the  abrasion  or  breaks  in  it. 


MATERIALS. 


195 


Zinc,  in  the  pure  form  of  spelter,  is  crystalline  and  brittle,  but,  at  a  temperature  be- 
tween 210°  and  300°,  it  is  so  ductile  and  malleable  that  it  can  be  readily  rolled  into  sheets, 
and  of  late  has  been  used  as  a  cheap  substitute  for  sheet-copper ;  but,  under  considerable 
variations  of  temperature,  as  for  lining  of  bath-tubs,  it  takes  permanent  wrinkles,  and,  for 
coverings  of  roofs,  suitable  provision  must  be  made  for  its  expansion.  But  as  a  plating  of 
iron,  under  the  name  of  galvanizing,  it  affords  un  admirable  protection,  cheaply,  and  ex- 
tends the  use  of  iron  in  sheets,  bolts,  and  castings,  where  it  would  not  otherwise  be  appli- 
cable. Zinc,  as  a  pigment,  does  not  discolor,  like  lead,  under  the  action  of  sulphureted 
hydrogen,  but  is  objected  to  by  painters  for  its  want  of  body  or  cover. 

METALS. 


METALS  AND  ALLOYS. 

Specific 
gravity. 

Weight 
per  c.  ft. 

Melting- 
point. 

Resistance  in  pounds  per  square  inch. 

To  crushing. 

To  tension.' 

Aluminum-bronze 

•  Fahr. 

73,000-96,000 
1,060 
3,250 
18,000 
22,000 

13,'000-25',000 
1,800 
22,000-50,000 
55,000 
40,000 
85,000-145,000 
4,600 
2,500 
40,000-  60,000 
70,000-120,000 
50,000-100,000 
120,000-200,000 
50,000-  85,000 

Antimony   cast 

4-500 
9-900 
8-500 
8-726 
19-238 
7'20 
11-479 

280 

617 
530 
537 
1,200 
450 
716 

932 
476 
1,873 

4,587 
5,237 
18,000 
594 

Bismuth  

Brass  

50,000-160,000 
117,000 

82,000-14'5,000 
7,000 

'Copper 

<rold 

Iron           

Lead  

Phosphor-bronze  . 

Platinum,  cast.    .              

21-500 
10-480 
7-800 
7-250 
7-215 
7-77 
7-85 

1,340 
654 
486 
450 
450 
485 
490 

3,080 
3,677 

442 

700 
2,822 
2,462 

Silver            " 

Steel              " 

125,000-295,000 
15,500 

Tin                 "... 

Zinc               "  

Iron    forced 

40,000-  65,000 
100,000-180,000 

Steel       " 

Iron  wire  (unannealed)    . 

Steel  wire           "           i     

Sterro-metal 

7 



Fig.  328  is  an  admirable  illustration  of  the  graphic  representation  of  facts 
adopted  by  Professor  R.  H.  Thurston  of  exhibiting  the  results  of  his  tests  on 
the  strength  of  alloys — which  not  only  exhibits  the  results,  but  enables  others 
to  judge  the  probable  strength  of  other  mixtures.  The  apices  of  the  triangle 
marked  copper,  tin,  and  zinc,  represent  the  points  of  pure  metal,  100  per  cent. 
The  lines  opposite  the  apex  of  any  metal  represent  the  0  of  such  metal — thus  the 
base  opposite  copper  represents  an  alloy  of  tin  and  zinc  only,  without  any  cop- 
per, and  every  line  drawn  above  this  base,  and  parallel  to  it,  will  contain  a  per- 
centage of  copper  increasing  by  regular  scale,  from  the  base  to  the  apex,  and  so 
with  lines  opposite  tin  and  zinc  ;  the  first  contains  only  copper  and  zinc,  the 
latter  tin  and  copper,  and  the  percentages  of  tin  and  zinc  increase  with  the 
distance  from  their  opposite  lines  to  their  vertices.  It  will  be  seen  that  the 
intersections  of  these  percentage  parallels  define  the  percentages  of  each  metal, 
their  sum  always  making  100  per  cent.  If,  then,  the  strength  of  such  alloy, 
as  obtained  by  test,  be  supposed  to  represent  an  ordinate  or  elevation,  on  any 
convenient  scale,  and  be  represented  by  this  height  at  its  opposite  intersection 
of  percentage,  a  contour  map,  as  in  the  figure,  may  be  formed — which  the  pro- 
fessor has  not  only  done,  but  made  a  model  from  it.  The  summit,  65,000  on 
the  figure,  represents  the  position  of  the  strongest  alloy  found :  if  through  the 


196 


MATERIALS. 


scales  marked  copper  on  each  side,  we  find  the  parallel  to  the  base,  which  passes 
through  this  summit,  it  will  be  found  to  be  about  55,  that  is,  55  per  cent  cop- 
per. In  like  manner,  the  parallel  to  the  o  zinc  base,  intersecting  this  summit, 
Avill  be  about  43  per  cent  zinc ;  and,  in  the  same  way,  tin  is  2  per  cent.  If  we 


<VZ± 


-% 


FIG.  328. 

wish  to  find  the  probable  strength  of  any  mixture,  it  is  only  necessary  to  find 
the  contour  intersected  by  the  triple  parallels  representing  the  percentages 
which  we  are  investigating.  It  is  said  probable  strength,  because  the  care  and 
manipulation  of  the  founder  are  such  important  factors  in  the  result. 

Sulphur,  when  used  in  sufficiently  large  masses  as  to  show  on  a  drawing,  may  be  repre- 
sented by  a  reddish-yellow  tint,  or  some  distinctive  hatching.  It  melts  at  248°  Fahr.,  and, 
from  its  fluidity,  answers  admirably  for  the  filling  of  joints  between  stones,  beneath  the 
balls  of  iron  columns,  between  wood  and  stone,  and  around  anchor-bolts  in  stone,  forming, 
when  cold,  a  strong,  uniform  bearing,  and  adapting  itself  to  the  roughness  of  the  material, 
and  is  detached  with  difficulty.  It  is  used  largely  for  the  bases  of  engines,  and  for  the 
joints  of  the  cap-stones  of  dams.  On  the  dam  across  the  Mohawk,  at  Cohoes,  many  tons 
were  used  in  these  joints,  the  depth  of  sulphur  being  about  6  inches,  and  now,  after 
seventeen  years'  use,  but  few  of  the  joints  are  little  worn,  and  there  has  been  no  injurious 
effect  from  the  sulphur  on  the  limestone,  of  which  the  apron  or  capping  is  composed.  It 
is  better  for  most  of  the  above  purposes  than  lead,  being  cheaper,  more  fluid  when  molten,. 


MATERIALS. 


197 


shrinks  less  in  cooling,  is  less  affected  by  temperature,  and  its  crushing  strength  is  adequate 
to  any  of  the  positions  of  use  above,  but  it  is  brittle  under  blows.  It  sometimes  rusts  the 
bolts  or  iron  with  which  it  is  brought  in  contact,  but  this  is  prevented  by  an  addition  of 
about  20  per  cent  of  coal  tar.  This  mixture  is  used  as  a  cement  to  fasten  lights  in  illumi- 
nated tile  and  vault  covers. 

When  heated  to  about  300°,  sulphur  begins  to  grow  viscid,  and  at  428°  it  has  the  con- 
sistency of  thick  molasses.  Above  this,  it  begins  to  grow  thin  again.  Heated  to  518°, 
and  thrown  into  cold  water,  it  becomes  for  a  time  plastic,  and  is  used  for  taking  molds  or 
casts. 

Sulphur,  in  powder,  mixed  in  proportions  of  one  sal-ammoniac,  two  sulphur,  and  fifty 
of  iron-filings,  makes  a  mastic  which  is  used  for  calking  the  joints  of  iron  pipes,  especially 
gas-pipes.  The  joint  is  called  a  rust-joint. 

Glass,  in  drawing,  is  represented  by  a  bluish  tint  or  by  different  shades 
or  hatchings,  expressive  of  the  effect  of  light  upon  it,  whether  the  light  is 
reflected  or  transmitted. 

Fig.  329  represents  a  portion  of  a  mirror 
when  the  light  is  reflected.  The  exterior  of 
windows  is  often  represented  in  the  same  way, 
but  with  deeper  shades,  and  often  with  a  piece 
of  curtain  behind  in  white  with  dim  outline. 
A  window  viewed  from  inside  is  represented  in 
shades  less  than  in  the  figure,  or  as  transpar- 
ent, which  is  conveyed  by  the  dimness  of  out- 
line of  figures  or  skies  seen  beyond. 

Fig.  330  represents  a  glass  flask. 

Fig.  331  represents  a  glass  box  with  glass 
sides. 

Fig.  332  represents  a  glass  jar  containing 
fluids  of  different  densities. 

Figs.  333  and  334  represent  spars,  which  may  be  taken  for  any  transparent 
substances,  as  glass,  ice,  and  the  like. 


FIG.  329. 


FIG.  330. 


FIG.  331. 


Common  window-glass  is  blown  in  the  form  of  cylin- 
ders (hence  called  cylinder-glass),  flatted  out,  and  cut 
in  lights  of  varying  dimensions,  from  6  x  8  up  to  30  x  30 
inches,  and  put  up  in  boxes  containing  about  fifty  square 
feet.  It  is  classed  as  single-thick  (about  TV  inch)  and 
double-thick  (-J-  inch).  When  the  squares  are  large,  or  used  for  sky-lights,  they  should  be 
the  latter.  Plate-glass—polished  plate  is  used  for  windows  of  stores  and  first-class  build- 


FIG. 


198  MATERIALS. 

ings.  It  can  be  got  of  almost  any  dimensions,  and  of  a  thickness  from  T\  to  f  of  an  inch. 
Rough  plate  is  largely  used  for  floor-lights  and  sky-lights.  It  is  cut  to  required  sizes,  and 
of  a  thickness  from  f  to  one  inch. 

Single  thick  cylinder-glass  cuts   off  from  about  8  to  15  per  cent  of  the  light. 

Double-cylinder,  from  12  to  20  per  cent  of  the  light. 

Polished  plate,  three  sixteenth  inch  thick,  from  5  to  7  per  cent  of  the  light. 

Rough  plate,  one  half  inch  thick,  from  20  to  30  per  cent  of  the  light. 

Rough  plate,  one  inch  thick,  from  30  to  40  per  cent  of  the  light. 

This  is  when  the  glass  is  clean  ;  but  there  is  always  a  film  of  moisture  on  its  surfacer 
which  attracts  dust,  and  impairs  very  much  the  transmitted  light.  Rough  plate  more 
readily  retains  the  dirt,  and,  when  it  is  used  as  floor-lights,  becomes  scratched.  It  is 
therefore  usual,  in  the  better  class  of  buildings,  to  use  a  cast  white  glass,  set  in  iron  frames. 
In  outer,  or  platform  lights,  these  lights  are  in  the  form  of  lenses,  set  in  cast-iron  frames, 
with  an  asphalt  putty,  or  resting  on  iron  frames  and  imbedded  in  Portland  cement. 


Fi&.  333.  FIG.  334. 

Rubber,  mixed  and  ground  with  sulphur,  subjected  to  heat,  becomes  vulcanized,  and  is 
not  affected  by  moderate  variations  in  temperature.  Soft  rubber,  most  extensively  used  for 
industrial  purposes,  is  subjected  to  a  heat  of  from  265°  to  300°,  and  for  a  time  can  withstand 
a  temperature  a  little  below  this  without  losing  its  elasticity;  after  a  time  it  will  harden. 
Soft  rubber  is  classed  as  pure  rubber,  and  fibrous  rubber,  or  rubber  with  cloth.  Pure 
rubber  contains  about  fifty  per  cent  of  rubber  and  fifty  per  cent  of  compound,  white  lead 
and  sulphur.  It  is  used  for  the  buffers  and  springs  of  railway-carriages,  and  for  the  faces 
of  valves  and  seats  of  water-pumps,  but  it  is  not  well  suited  for  the  pumping  of  hot  water, 
especially  above  212°,  as  it  is  liable  to  lose  its  elasticity ;  and,  although  some  valves  may 
stand  a  considerable  time,  it  is  almost  impossible  to  secure  uniformity  in  the  rubber. 
Fibrous  rubber — rubber  ground  with  cotton  or  other  fiber,  or  spread  on  cloth,  on  more  or 
less  thicknesses — is  used  for  the  packing  of  faced  joints  of  pipes  and  gaskets  for  water  or 
steam.  It  makes  a  stanch  joint,  and,  even  when  hardened  under  heat,  it  still  preserves 
it.  Rubber  cloth  is  also  used  for  belting  and  hose-pipes.  When  used  for  the  convey- 
ance of  steam,  the  inner  coat  is  the  first  affected,  and  it  may  be  some  time  before  the 
whole  pipe  suffers.  In  buying  rubber,  explain  the  purpose  to  which  it  is  to  be  appliedr 
and  depend  on  the  guarantee  of  the  vender.  Rubber  is  often  to  be  designated  by  the 
draughtsman,  which  it  may  be  by  a  bluish-black  tint,  or  by  lines  across  it  parallel  to  its 
length. 

Paints  are  used  for  a  twofold  purpose— for  covering  and  preserving  the  material  to 
which  they  are  applied,  and  for  ornamentation.  The  best  and  the  most  general  is  white-lead 
ground  with  linseed-oil,  either  used  by  itself  or  mixed  with  various  other  pigments,  a& 
ochre,  chrome,  lamp-black,  etc.  It  is  often  adulterated  with  barytes.  For  the  covering 
of  iron,  or  for  the  packing  of  close  joints  in  it,  nothing  is  better  than  pure  red-lead,  but 
many  of  the  oxides  of  iron,  red  or  yellow,  form  good  covers  of  iron,  and,  as  cheap  and 
good  paints,  are  used  on  tin  roofs.  All  the  leads  and  pigments  are  ground  in  oil :  if  the 
oil  is  raw,  it  dries  slowly ;  driers,  as  litharge,  are  added  to  hurry  the  process,  but,  with 


MATERIALS. 


199 


boiled  oil,  no  drier  is  necessary.  Almost  any  inert  substance,  as  cement,  chalk,  or  sand, 
if  fine  enough,  can  be  ground  with  oil  for  a  paint,  and  make  a  good  cover,  and  for  these 
fish-oil  will  answer.  The  general  specification  for  painting  is  "  paint  with  —  good  coats  of 
white-lead,  of  such  color  as  may  be  directed."  The  priming-coat  of  new  wood- work 
requires  more  oil  than  paint.  For  the  next  coats,  one-half  pound  of  paint  to  the  square  yard 
would  be  considered  a  good  coat.  If  the  paint  is  on  old  work,  or  that  which  has  been 
already  painted,  there  will  be  a  little  less  lead  required.  Wood  should  be  fairly  dry  before 
the  application  of  paint,  so  that  it  may  properly  adhere  and  not  inclose  moisture  that  may 
rot  the  wood.  The  knots  should  be  Trilled,  that  is,  covered  with  shellac  varnish  or  similar 
preparation,  to  prevent  the  exuding  of  the  resin.  The  heads  of  nails  should  be  sunk,  and 
the  holes  and  cracks  filled  with  putty,  and  the  surface  of  the  wood  smoothed. 

Coals  and  other  minerals  are  represented  like  rocks  or  stones,  in  varied  shades  of  tones 
and  colors.  Fig.  334a  represents  the  fire-box  of  a  locomotive,  with  coal  in  the  state  of 
ignition  in  its  usual  type.  In  color,  flame  is  represented  in  streaks  of  red-yellow,  with 
dark  tints  for  smoke.  Water  occupies  the  lower  half  of  the  boiler ;  but,  as  steam  under 


FIG. 


FIG.  3346. 


pressure  is  invisible  like  gas,  the  space  occupied  by  it  is  shown  as  empty.  If  the  direction 
of  its  movement  is  desired,  it  is  indicated  by  arrows.  Steam  issuing  into  atmosphere,  or 
boiling  in  an  open  kettle,  has  the  appearance  of  a  very  light  smoke  or  cloud  (Fig.  3345). 

There  are  many  substances  used  in  such  masses  in  construction,  or  to  be  shown  in  the 
processes  of  manufacture,  that  must  be  graphically  represented  by  the  draughtsman  by  a 
general  imitation  of  their  natural  appearance,  or  conventionally  with  explanatory  marginal 
blocks  and  legends. 


MECHANICS. 


THE  draughtsman,  in  designing  a  structure,  should  be  conversant  not  only  with  the 
nature  of  the  material,  but  also  with  the  forces  to  which  it  is  to  be  subjected — their  mag- 
nitude, direction,  and  points  of  application,  and  their  effects;  that  is,  he  should  know  the 
iirst  principles  of  mechanics,  the  science  of  rest,  motion,  and  force — to  wit,  Statics,  Dynam- 
ics, and  Kinematics.  Statics  treats  of  balanced  forces,  or  rest ;  dynamics,  of  unbalanced 
forces,  where  motion  ensues ;  and  kinematics,  of  the  comparison  of  motions  with  each 
other.  Considering  statical  forces  simply  in  the  abstract,  the  bodies  to  which  they  are 
applied  are  assumed  as  perfectly  rigid,  without  breaking,  binding,  twisting,  or  in  any  wise 
changing  by  the  application  of  such  forces. 

Force  is  a  cause  tending  to  change  the  condition  of  a  body  as  to  rest  or  motion.  Force 
is  measured  by  weight.  In  England  and  the  United  States  the  unit  of  force  is  the  pound, 
on  the  Continent  the  gramme.  All  bodies  fall,  or  tend  to  fall,  to  the  earth.  This  force  is 
called  the  attraction  of  gravitation.  Its  direction  is  shown  by  that  of  a  string  from 
which  a  weight  is  suspended  (Fig.  335).  It  is  called  a  vertical  line,  and  its  direction  is 
toward  the  center  of  the  earth.  Practically,  these  lines  are  considered 
parallels.  Let  a  mass,  P  (Fig.  336),  be  suspended  by  a  cord.  Each  particle 
is  acted  on  by  gravity,  and  the  resultant  of  all  these  parallel  forces  is  the 
force  resisted  by  the  cord,  or  the  entire  weight  of  the  body.  If  a  mass 
(Fig.  337)  be  suspended  from  two  different  points,  P  and  Q,  the  directions 
of  the  string  will  meet  at  a  point  C,  which  is  called  the  center  of  gravity, 
where  all  the  weight  may  be  considered  to  be  concentrated.  When  a  body 
of  uniform  density  has  a  center  of  symmetry  (a  point  which  bisects  all 
straight  lines  drawn  through  it),  this  point  coincides  with  the  center  of 


FIG.  335. 


FIG.  336. 


FIG.  337. 


FIG.  338. 


gravity,  as  the  middle  of  a  straight  line,  the  center  of  a  circle,  the  intersection  of  the 
diagonals  of  a  parallelogram,  the  intersection  of  lines  drawn  from  any  two  angles  of  a 
triangle  to  the  centers  of  the  opposite  sides ;  in  solids,  the  center  of  a  sphere,  the  middle 
point  of  the  axis  of  a  cylinder,  and  the  intersection  of  the  diagonals  of  a  parallelepiped. 

The  center  of  gravity  of  the  triangular  pyramid,  Fig.  338,  is  in  the  straight  line  A  E, 
connecting  the  apex  A  with  the  center  of  gravity  of  the  base  triangle  BCD,  and  distant 
i  of  the  length  of  the  line  A  E  from  E. 


MECHANICS. 


201 


The  center  of  gravity  of  solids,  which  may  be  divided  into  symmetrical  figures  and 
pyramids,  as  for  all  practical  purposes  most  may  be,  can  be  found  by  determining  the 
center  of  gravity  of  each  of  the  solids  of  which  it  is  compounded,  and  then  compound- 
ing them,  observing  that  each  center  of  gravity  represents  the  solid  contents  of  its  own 
mass  or  masses  of  which  it  may  be  composed.  The  center  of  gravity  of  bodies  enclosed 
by  more  or  less  regular  contours,  as  a  ship  for  instance,  is  determined  by  dividing  it  into 
parallel  and  equidistant  sections,  finding  the  center  of  gravity  of  each,  and  compounding 
them  into  a  single  one. 

The  center  of  gravity  of  a  body  may  be  determined  practically,  as  shown  above,  by  its 
suspension  from  different  points.  It  can  be  done  generally  more  readily  by  balancing  the 
body  in  horizontal  positions  on  different  lines  of  support ;  the  center  of  gravity  will  lie 
in  the  intersection  of  planes  perpendicular  to  these  lines.  A  body  placed  in  a  horizontal 
position  will  fall  over,  unless  the  vertical  line  from  the  center  of  gravity  falls  within  the 


FIG.  339. 


FIG.  340. 


FIG.  341. 


FIG.  342. 


base  of  support ;  as  Fig.  339  will  stand,  while  Fig.  340  will  fall  over.  A  person  car- 
rying a  weight  insensibly  throws  a  portion  of  the  body  forward,  backward,  or  laterally,  to 
balance  the  load.  Thus,  in  Fig.  341,  the  body  is  thrown  back,  so  that  the  vertical  from 
the  center  of  gravity  ^,  compounded  of  the  center  of  gravity  G  of  the  woman  and  of  the 
load  H,  falls  within  the  base  of  the  feet. 

When  a  figure  rests  in  such  a  position  that  its  center  of  gravity  is  in  its  lowest  position, 
it  is  said  to  be  in  stable  equilibrium.  It  may,  like  a  ball,  rest  in  any  position,  as  the  center 
of  gravity  is  neither  depressed  nor  raised  by  movement ;  but,  in  the  ellipsoidal  form  (Fig. 


FIG.  343. 


FIG.  344. 


FIG.  345. 


342)  or  in  the  toy  (Fig.  343),  any  movement  tends  to  raise  the 
center  of  gravity,  and,  on  the  cessation  of  the  force,  the  body 
returns  to  its  original  position.  The  ellipsoidal  form  (Fig.  344), 
placed  on  its  pointed  end,  is  balanced,  but  the  slightest  move- 
ment lowers  the  center  of  gravity,  and,  without  the  applica- 
tion of  an  outside  force,  it  can  not  be  raised,  and  therefore  falls.  This  is  called  unstable 
equilibrium.  In  the  toy  (Fig.  345),  the  body  of  the  figure  is  light,  and  the  weight  of  the 
balls  brings  the  center  below  the  point  of  support.  This  will  admit  of  great  oscillation, 
and  return  to  its  original  position.  A  cork  with  two  forks  inserted  in  it,  like  the  wires 
of  the  balls,  and  resting  on  the  top  of  a  glass,  will  illustrate  this  readily. 


202 


MECHANICS. 


FIG.  346. 


When  two  parallel  forces,  F  F',  are  applied  at  the  extremities  of  a  straight  line  (Fig.  346), 
they  have  a  resultant,  K,  equal  to  their  sum,  and  acting  at  a  point,  0,  which  divides  the  line 
inversely  proportional  to  the  forces.  If  the  forces  are  equal, 
the  point  0  will  be  at  the  center  of  the  line ;  if  the  force  F  is 
double  that  of  F',  C  A  will  be  equal  to  one  half  C  B.  This  is 
called  the  principle  of  the  lever. 

Levers,  in  practice,  are  called  of  the  first  (Fig.  347),  second 
(Fig.  348),  and  third  class  (Fig.  349),  according  to  the  position, 
weight,  W,  power  applied,  P,  and  fulcrum,  support  or  turning- 
point,  C,  of  the  lever.  They  are  all  forces,  and  only  vary  in 
name.  The  two  extreme  forces  must  always  act  in  the  same 
direction ;  the  middle  one  must  act  in  the  opposite  direction,  and  be  equal  to  the  sum 
of  the  other  two;  and  the  magnitude  of  the  extreme  forces  be  Diversely  proportional  to 
their  distances  from  the  middle  one.  Let  the  middle  force  C  be  measured  by  a  spring- 
balance  (Fig.  350) ;  it  will  mark  the  sum  of  the 
weights  a  and  5.  Call  the  distance  from  a  to  c,  #, 
and  from  5  to  c,  y,  then  the  weight  a  will  be  to 
the  weight  5  as  y  is  to  35,  or  a  x  =  5  y.  Suppose  the 
weight  a  to  be  6  pounds  and  at  5  3  pounds,  at  c  it 


FIG.  347. 


W    F 


FIG.  348. 


a 


FIG.  350. 


.Q 


FIG.  349. 


FIG.  351. 


will  be  9  pounds,  and  a  c  or  x  will  be  to  &  c  or  y  as  6  to  3,  or,  if  the  lever  is  48  inches, 
&  c  will  be  16  inches  and  ac  32  inches. 

To  find  graphically  the  fulcrum,  or  point,  at  which  a  lever  should  be  sup- 
ported to  sustain  in  equilibrium  weights,  or  equivalent  forces,  acting  at  the 
extremities  of  the  lever.  Let  A  B  (Fig.  351)  be  the  lever.  At  A  and  B  let 
fall  and  erect  perpendiculars  to  the  lever.  Lay  off  from  A,  on  any  con- 
venient scale,  A  B',  corresponding  to  the  weight  applied  at  B  ;  and  at  B,  on 
the  same  scale,  B  A',  the  weight  applied  at  A  ;  draw  the  line  A'  B' ;  its  inter- 


MECHANICS. 


203 


F' 


™ 


FIG.  352. 


section,  F,  with  the  lever  will  be  the  position  of  the  f ulci^^/^  T^his  is  on  the 
hypothesis  that  there  is  no  weight  to  the  lever,  or  that,  after  determining  the 
position  of  the  fulcrum,  the  lever  itself  is  balanced  on  the  point  by  the  addi- 
tion of  weight  on  the  short  arm  F  A,  or  the  reduction  of  weight  on  the  long 
one  F  B.  If  the  lever  is  of  uniform  weight, 
on  perpendiculars  to  C,  the  center  of  the 
lever  (Fig.  352),  and  to  F,  the  fulcrum,  as 
before  determined,  lay  off  F  C',  the  weight 
of  the  lever,  and  C  F',  the  sum  of  the 
weights  applied  at  A  and  B  ;  draw  C'  F'. 
Its  intersection,  F",  will  be  the  actual  ful- 
crum, taking  into  consideration  the  weight 
of  the  lever  in  addition  to  the  weights  sus- 
pended at  the  extremities. 

The  Wheel  and  Axle. — If  a  weight,  P,  be  sus- 
pended from  the  periphery  of  a  wheel  (Fig.  353), 
while  another  weight,  W,  is  suspended  on  the  op- 
posite side  of  a  barrel  or  axle  attached  to  the 
wheel,  the  principle  of  action  is  the  same  as  that 
of  the  lever.  P  multiplied  by  its  length  of  lever 
or  radius  ca  of  the  wheel  is  equal  to  W  multiplied  by  its  length  of  lever  or  radius 
of  the  axle  cb ;  the  axis  c  is  the  fulcrum.  If  a  movement  downward  be  communicated 
to  P,  as  shown  by  the  dotted  line,  a  rotary  motion  is  given  to  the  wheel  and  axle;  the 
cord  of  P  is  unwound  while  that  of  W  is  wound  up,  but  P  is 
still  suspended  from  a  and  W  from  & ;  the  leverage,  or  dis- 
tance from  the  fulcrum,  of  each  is  the  same  as  at  first.  The 
wheel  and  axle  is  a  lever  of  continuous  and  uniform  action. 
Since  the  wheel  has  a  larger  circumference  than  the  axle,  by 
their  revolution  more  cord  will  he  unwound  from  the  former 
than  is  wound  up  on  the  latter,  P  will  descend  faster  than  TV 
is  raised,  in  the  proportion  of  the  circumference  of  the  wheel 
to  that  of  the  axle,  or  of  their  radii  ca  to  c  &.  When  P  has 
reached  the  position  P',  W  will  have  reached  W.  If  c  a  be 
four  times  c  5,  then  P  will  have  moved  four  times  the  dis- 
tance that  W  has.  The  movement  is  directly  as  the  length 
of  the  levers,  or  the  radii  of  the  points  of  suspension.  It 
will  be  perceived,  therefore,  to  move  a  large  weight  by  the 
means  of  a  smaller  one,  that  the  smaller  must  move  through 
the  most  space,  and  that  the  spaces  described  are  as  the  op- 
posite ends  of  the  lever,  or  inversely  as  the  weights. 

It  is  the  fundamental  principle  of  the  action  of  all  me- 
chanical powers,  that  whatever  is  "  gained  in  power,"  as  it 
is  said,  is  lost  in  space  traveled;  that,  if  a  weight  is  to  be 
raised  a  certain  number  of  feet,  the  force  exerted  to  do 
this  must  always  be  equal  to  the  product  of  the  weight 
by  the  height  to  which  it  is  to  be  raised  ;  thus,  if  200 
pounds  are  to  be  raised  50  feet,  the  force  exerted  to  do  this 
must  be  equal  to  a  weight,  which,  if  multiplied  by  its  fall, 
will  be  equal  to  the  product  200  x  50,  or  10,000 ;  and  it  is  immaterial  whether  the 
force  be  a  weight  of  10,000  pounds  falling  1  foot,  or  1  pound  10,000  feet. 


FIG.  353. 


204 


MECHANICS. 


It  is  now  common  to  refer  all  forces  exerted  to  a  unit  of  pounds-feet,  that  is,  1  pound 
falling  1  foot ;  and  the  effect  to  the  same  unit  of  pounds-feet,  1  pound  raised  1  foot. 
Thus,  in  the  example  above,  the  force  exerted  or  power  is  10,000  pounds-feet  falling  ;  the 
effect  10,000  pounds-feet  raised.  In  practice,  the  pounds-feet  of  force  exerted  must  always 
be  more  than  the  pounds-feet  of  effect  produced  ;  that  is,  there  must  be  some  excess  of 
the  former  to  produce  movement,  and  to  overcome  resistance  and  friction  of  parts. 

The  measure  of  any  force,  as  represented  by  falling  weight,  is  termed  the  absolute  power 
of  that  force  ;  the  resulting  force,  or  useful  effect  for  the  purposes  for  which  it  is  applied,  is 
called  the  effective  power. 

The  Pulley.— The  single  fixed  pulley  (Fig.  354)  consists  of  a  single  grooved  wheel 
movable  on  a  pin  or  axis,  called  fixed,  because  the  strap  through  which  the  pin  passes  is 
attached  to  some  fixed  object.  A  rope  passes  over  the  wheel  in  the  groove;  on  one  side 
the  force  is  exerted,  and  on  the  other  the  weight  is  attached  and  raised.  It  may  be  con- 
sidered a  wheel  and  axle  of  equal  diameters,  or  as  a  lever  in  which  the  two  sides  are  equal, 
the  pin  being  the  fulcrum.  P,  the  force  exerted,  must  therefore  be  equal  to  the  weight 
"W,  raised  ;  and,  if  movement  takes  place,  W  will  rise  as  much  as  P  descends. 

The  fixed  pulley  is  used  for  its  convenience  in  the  application  of  the  force ;  it  may  be 
easier  to  pull  down  than  up,  for  instance  ;  but  the  pounds  of  force  must  be  equal  to  the 
pounds  of  effect.  The  tension  on  the  rope  is  equal  to  either  the  force  or  weight. 

Fig.  355  is  a  combination  of  a  fixed  pulley,  A,  and  a  movable  pulley,  B.     The  simplest 
way  to  arrive  at  the  principle  of  this  combination  is  to  consider  its  action.    Let  P  be  pulled 
down,  say  two  feet;  the  length  of  rope  drawn  to  this  side  of  the  pulley  must  be  furnished 
from  the  opposite  side.     On  that  side  there  is  a  loop,  in  which  the  movable 
pulley,  with  the  weight  W  attached,  is  suspended.     Each  side  of  this  loop,  2 


and  3,  must  go  to  make  up  the  two  feet  for  the  side  or  end  1. 
will  therefore  furnish  each  one  foot.     As  these  cords  are 
shortened  one  foot,  the  weight  W  is  raised  one  foot,  and,  as 


Cords  2  and  3 


FIG.  354. 


FIG.  355. 


FIG.  356. 


FIG.  357. 


the  movement  of  W  is  but  one  foot  for  the  two  feet  of  P,  W  must  be  twice  that  of  P, 
because  the  two  pounds-feet  of  P  must  equal  the  pounds-feet  of  W. 

In  the  combination  of  pulleys  (Fig.  356),  let  P  be  pulled,  say  three  feet;  then  this 
length  of  rope,  drawn  from  the  opposite  side  of  the  pulley,  is  distributed  over  the  three 
cords,  2,  3,  4,  and  the  weight  W  is  raised  one  foot ;  consequently,  the  weight  W  is  three 
times  that  of  P.  The  cord  1  supports  P,  the  cords  2,  3,  4,  the  weight  W,  or  three  times 
P;  consequently,  the  tension  on  every  cord  is  alike.  The  same  rope  passing  freely  around 
pulleys  must  have  the  same  tension  throughout ;  so  that,  to  determine  the  relation  of  W 
to  P,  count  the  number  of  cords  which  sustain  the  weight.  Thus,  in  Fig.  357,  the  weight 
is  sustained  by  four  cords ;  consequently,  it  is  four  times  the  tension  of  the  cord,  or  four 
times  the  force  P.  In  order  not  to  confuse  the  cords,  the  pulleys  are  represented  as  in  the 
figures ;  but,  in  construction,  the  pulleys,  or  sheaves,  are  usually  of  the  same  diameter, 
and  those  in  connection,  as  A  and  B,  and  C  and  D,  run  on  the  same  pin. 


MECHANICS. 


205 


The  Inclined  Plane. — To  support  a  weight  by  means  of  a  single  fixed  pulley,  the  force 
must  be  equal  to  the  weight.  Suppose  the  weight,  instead  of  hanging  freely,  to  rest  upon 
an  inclined  plane  b  d  (Fig.  358)  ;  if  motion  ensue,  to  raise  the  weight  W  the  height  a  5,  the 
rope  transferred  from  the  weight  side  of  the  pulley  will  be  equal  to  5  d,  and  P  will  have, 
consequently,  fallen  this  amount ;  thus,  if  b  d  be  six  feet,  and  a  ft  one  foot,  while  W  is  raised 
one  foot,  P  has  descended  six  feet,  and,  as  pounds-feet  of  power  must  equal  pounds-feet  of 
effect,  P  will  be  one  sixth  of  W  ;  and,  by  reference  to  the  figure,  P  is  to  W  as  a  5  is  to  5  d, 
or  as  the  height  of  the  incline  is  to  its  length.  If  the  end  of  the  plane  d  be  raised,  till  it 
becomes  horizontal,  the  whole  weight  would  rest  on  the  plane,  and  no  force  would  be 
necessary  at  P  to  keep  it  in  position;  if  the  plane  be  revolved  on  5,  till  it  becomes  per- 


FIG.  358. 


FIG.  359. 


pendicular,  then  the  weight  is  not  supported  by  the  plane  at  all,  but  it  is  wholly  depen- 
dent on  the  force  P,  and  is  equal  to  it.  Between  the  limits,  therefore,  of  a  level  and  a 
perpendicular  plane,  to  support  a  given  weight  W,  the  force  P  varies  from  nothing  to  an 
equality  with  the  weight. 

The  construction  (Fig.  359)  illustrates  the  principle  of  the  wedge,  which  is  but  a  mova- 
ble inclined  plane ;  if  the  wedge  be  drawn  forward  by  the  weight  P,  and  the  weight 
"W  be  kept  from  sliding  laterally,  the  fall  of  P  a  distance  equal  to  a  d  will  raise  the 
weight  W  a  height  cl.  P  will  therefore  be  to  W  as  c  5  is  to  a  d.  For  example,  if  the 
length  of  the  wedge  a  d  be  ten  feet,  and  the  back  c  ~b  two  feet,  then  P  will  be  to  W  as 
two  to  ten,  or  one  fifth  of  it. 

Let  the  inclined  plane  a  &  d  (Fig.  359)  be  bent  round,  and  attached  to  the  drum  A 
(Fig.  360),  to  which  motion  of  revolution  on  its  axis  is  given,  by  the  unwinding  of  the 
turns  of  a  cord  from  around  its  periphery,  through  the  action  of  a  weight  P  suspended 

from  a  cord  passing  over  a  pulley.  If  the  weight  W 
be  retained  in  its  vertical  position,  by  the  revolution 
of  the  drum,  it  will  be  forced  up  the  incline,  and 
when  the  cord  has  unwound  one  half  turn  from  the 
drum,  and  consequently  the  weight  P  descended  a 

distance,  c  e,  equal  to 
one  half  the  circum- 
ference of  the  drum, 
the  weight  W  has  been 
raised  to  the  height  a  & 
by  the  half  revolution 
of  the  plane ;  P  must 
therefore  be  to  W  as 
a  5  is  to  one  half  the 
circumference.  Extend 
the  inclined  plane  so  as 
to  encircle  the  drum  (Fig.  361).  The  figure  illustrates  the  mechanism  of  the  screw,  which 
may  be  considered  as  formed  by  wrapping  a  fillet-band  or  thread  around  a  cylinder  at  a 
uniform  inclination  to  the  axis.  In  practice,  the  screw  or  nut,  as  the  case  may  be,  is  moved 
by  means  of  a  force  applied  at  the  extremity  of  a  lever,  a  complete  revolution  raises  the 


FIG.  360. 


FIG.  361. 


206 


MECHANICS. 


weight  the  distance  from  the  top  of  one  thread  to  the  top  of  the  one  above,  or  the  pitch. 
If  the  force  be  always  exerted  at  right  angles  to  the  lever  (Fig.  362),  the  lever  may  be  con- 
sidered the  radius  of  a  wheel,  at  the  circumference  of  which  the  force  is  applied.  Thus, 
if  the  lever  be  three  feet  long,  the  diameter  of  the  circle  would  be  six  feet,  and  the  cir- 


FIG.  362. 

cumference  6  x  3-1416,  or  18T8/o-  feet ;  if  the  pitch  be  one  inch,  or  one  twelfth  of  a  foot, 
then  the  force  would  be  to  the  weight  as  one  twelfth  is  to  18-85  ;  and  if  the  force  be  one 
pound,  the  weight  would  be  226*20  pounds. 

The  resultant  of  two  forces  of  exertion,  as  has  been  seen,  is  their  sum,  and  counter- 
balances the  forc^  of  resistance,  which  must  be  applied  at  a  point  intermediate  between, 
and  distant  from  each  of  them,  inversely  as  the  forces  exerted. 

The  resultant  of  any  number  of  parallel  forces  acting  in  one  direction  is  equal  to 
their  sum  acting  in  the  same  direction  at  some  intermediate  point ;  that  is,  the  effect  of 
all  these  forces  is  just  the  same  as  if  there  were  but  one  force,  equal  to  their  sum, 
acting  at  this  point,  and  is  balanced  by  an  equal  force  acting  in  the  opposite  direction. 
This  central  point  may  be  determined  by  finding  the  resultant,  i.  e.,  the  sum,  and  the 
point  of  application  for  any  two  of  the  forces,  as  shown  graphically  in  Figs.  351,  352,  and 
then  of  other  two,  the  resultants  thus  determined  being  again  added  together  like  simple 
forces. 

Inclined  Forces  are  those  whose  directions  are  inclined  to  each  other.  When  two 
men  of  equal  strength  pull  directly  opposite  to  each  other,  the  resultant  is  nothing.  Let 
a  third  take  hold  of  the  center  of  the  rope  (Fig.  363),  and  pull  at  right  angles  to  the 

rope;  he  will  make  an  angle  in  the  rope,  and  the 
other  two  now  pull  in  directions  inclined  to  each 
other.  The  less  the  force  exerted  at  the  center,  the 
less  the  flexure  in  the  rope ;  but  when  it  becomes 
equal  to  the  sum  of  the  forces  at  the  ends,  the  two, 
to  balance  it,  must  pull  directly  against  it,  bringing 
the  ends  of  the  rope  together,  and  acting  as  parallel 
forces.^  Between  the  smallest  force  and  the  largest  that  can  be  exerted  at  the  center  and 
maintain  a  balance  or  equilibrium,  the  ends  of  the  rope  assume  all  varieties  of  angles, 
which  angles  bear  definite  relations  to  the  forces. 

Represent  these  forces  by  weights  (Fig.  364).  Let  P  and  P'  be  the  extreme  forces  act- 
ing over  the  pulleys  M  and  N,  and  tending  to  draw  the  rope  straight,  which  the  weight  P" 
prevents.  Lay  off  the  weight  of  P  (90  pounds)  along  A  B,  and  the  weight  of  P'  (60  pounds) 
along  A  0.  Draw  En  parallel  to  A  C,  and  Cn  parallel  to  A  B.  Connect  n  with  A.  If 
this  is  measured  with  the  same  scale  that  A  B  and  A  0  were  laid  off  with,  it  will  be  found 
that  it  equals  120  pounds,  which  will  be  found  to  be  the  same  as  the  weight  P".  An,  there- 


FIG.  363. 


MECHANICS. 


207 


fore,  gives  the  amount  and  direction  of  the  resultant  of  the  two  forces  P  and  P',  which 
resultant  is  balanced  by  P".     In  the  same  way  the  resultant  of  any  number  of  inclined 


forces  (Fig.  365)  may  be  found  by  compounding  the  resultant  of  any  two  forces  with  a 
third,  and  so  on. 

As  two  forces  may  be  compounded  into  a  single  resultant,  so  conversely  one  force  may 
be  resolved  into  two  components ;  thus,  let  the  weight  P  (Fig.  366)  be  supported  by  two 
inclined  rafters,  C  A  and  C  B. 
Each  resists  a  part  of  the  force 
exerted  by  the  weight  P.  To 
find  the  force  exerted  against 
the  abutments  A  and  B,  in  the 
direction  of  C  A  and  C  B,  draw 
c  A'  (Fig.  367)  parallel  to  C  A, 
c  B'  to  C  B,  and  c  d,  a  paral- 
lel to  the  line  C  P,  the  direc- 
tion in  which  the  weight  P 
acts ;  lay  off.  c  d  from  a  scale 
of  equal  parts,  a  length  which 


FIG.  367. 


will  represent  the  number  of  pounds,  or  whatever  unit  of  weight  there  may  be  in  the 
weight  P ;  draw  d  a  parallel  to  c  B',  and  d  5  parallel  to  c  A' ;  c  a,  measured  on  the  scale 


208  MECHANICS. 

of  equal  parts  adopted,  will  represent  the  pounds  or  units  of  weight  exerted  against  A 
in  the  direction  of  0  A,  and  c  b  the  pounds  or  units  of  weight  exerted  against  B  in  the 
direction  of  C  B. 

This  method  of  finding  the  resultant  of  two  forces,  or  the  components  of  one  force,  is 
called  the  parallelogram  of  forces.  If  two  sides  of  a  parallelogram  represent  two  forces  in 
magnitude  and  direction,  the  resultant  of  these  two  forces  will  be  represented  in  magni- 
tude and  direction  by  the  diagonal  of  the  parallelogram  and  conversely. 

The  sum  of  ac  and  c  &  is  greater  than  c  d  ;  that  is,  the  weight  P  exerts  a  greater  force 
in  the  direction  of  the  lines  C  A  and  C  B,  against  A  and  B,  than  its  own  weight ;  but  the 
down  pressure  upon  A  and  B  is  only  equal  to  the  weight  of  P  and  of  the  rafters  which 
support  it,  which  last,  in  the  present  consideration,  is  neglected.  Ptesolve  c  5,  the  force 
acting  on  B  in  the  direction  of  eB',  into  g  ~b  or  ce  the  downward  pressure,  and  eg  or  eb 
the  horizontal  thrust  on  the  abutment  B,  and  ca  into  cf&ndfa.  To  decompose  a  force, 

form  a  triangle,  with  the  direction  of  the  other 
forces,  upon  the  line  representing  the  magnitude 
and  direction  of  the  given  force ;  c  e  represents 
the  weight  on  B,  c  f  the  weight  on  A  ;  c  d,  or 
c  e  +  d  e,  the  whole  weight  P ;  therefore,  the 
weight  upon  the  two  abutments  A  and  B  is 
equal  to  the  whole  weight  of  P. 

The  steelyard  (Fig.  368)  is  a  lever,  from  the 
short  arm  of  which  a  dependent  hook  or  scale 
supports  the  article  to  be  weighed  ;  while,  on 
the  long  arm,  a  fixed  weight,  P,  is  slid  in  or 

out  from  the  fulcrum  till  it  balances  the  article ;  the  distance  as  marked  on  a  scale  on  the 
long  arm  determines  the  weight.  In  platform-scales,  when  very  heavy  weights  are  bal- 
anced by  small  weights  on  a  graduated  arm,  combinations  of  levers  are  used,  the  principle 
of  which  can  be  understood  from  Fig.  369.  Thus,  suppose  PF  to  be  7",  a  F  2",  a  F'  9"r 
&F'2",  &F"11",  F"W  3". 

P  is  to  force  a  as  a  F    to    P  F,  or  as  2  to    7 

Force  a  is  to  6  as   b  F'  to  a  F,    or  as  2  to     9 

&  is  to  W  as  F"  W  to  b  F",  or  as  3  to  11 

P  is  to  W  as  12  to  693 

The  differential  axle,  or  Chinese  capstan,  consists  of  an  axle  with  two  different  diameters 
(Fig.  370),  the  weight  W  being  suspended  in  the  loop  of  a  cord  wound  around  these  axles 
in  opposite  directions  by  a  single  turn  of  the  axle.  The  weight  is  only  raised  or  low- 


FIG.  369. 

ered  by  the  difference  between  these  two  circumferences ;  one  takes  up  while  the  other 
lets  out,  and  the  P,  to  balance  W,  must  be  as  these  differences  of  circumference  of  axles 
is  to  the  circumference  of  the  wheel  from  which  P  is  suspended. 

The  differential  screw  (Fig.  371)  consists  of  an  exterior  screw,  A,  and  an  interior  screw, 
B.  By  the  revolution  of  the  arm,  the  screw  A  is  moved  through  the  plate  D  in  propor- 
tion to  its  pitch,  but  the  interior  screw  B  moves  inward  its  pitch,  and  the  movement  of 
W  is  only  the  pitch  of  A  less  th#t  of  B,  and  the  power  applied  is  to  the  weight  moved  as 
the  difference  of  these  pitches  is  to  the  circumference  described  by  the  power. 


MECHANICS. 


FIG.  370. 


FIG.  371. 


As  the  lever  (Fig.  372)  moves  under  the  action  of  power  or  weight,  the  lever  be- 
comes inclined  to  the  direction  of  the  forces,  but  the  forces  are  still  parallel.  The  rela- 
tions of  the  forces  to  each  other  are  not  changed,  but  the  absolute  action  of  each  is  only 


FIG.  373. 

that  due  to  the  length  a  5  and  5  c,  to  which  the  directions  of  the  forces  are  perpendicu- 
lar.     In  the  bent  levers  (Figs.  373  and  374)  the  action  of  the  forces  is  estimated-  on 

lengths  of  arms,  determined  by  the 
perpendiculars  a  b  and  b  c  let  fall 
from  the  fulcrum  on  the  directions 
of  the  forces. 

The  toggle-joint  (Fig.  375)  is 
much  used  for  presses.  The  force 
is  exerted  in  the  direction  of  the 
arrow  at  0,  and  the  effective  force 


o 


FIG.  874. 


14 


FIG.  375. 


210 


MECHANICS. 


is  to  separate  the  plates  A  and  B.  The  action  is  as  shown  in  Fig.  376.  Equal  movements, 
as  0-1,  1-2,  2-3,  correspond  to  unequal  movements  at  A  and  B.  as  A  a',  a'  a?,  a?  a3.  The 
nearer  the  force  C  is  to  the  line  A  B,  the  less  the  movement  a2  a3 ;  and,  consequently,  the 
force  C  exerts  greater  effects  in  intensity,  but  the  latter  is  less  in  movement. 

C 


lib. 


FIG.  376. 

Fig.  377  exhibits  the  principle  of  the  hydraulic  press.  The  small  plunger  or  piston  may 
be  considered  the  application  of  the  force,  and  the  large  one  the  weight  to  be  raised  to 
balance  each  other ;  the  pressure  per  square  inch  of  surface  must  be  the  same,  and  the 
force  must  be  to  weight  as  the  surface  of  its  piston  is  to  that  of  the  weight-piston.  If 
16  j.  motion  takes  place,  the  force  will  move  through  space  cor- 
responding to  the  area  of  weight-piston,  while  the  weight 
will  move  that  of  the  area  of  the  force-piston.  And  this  is 
the  great  principle  of  all  mechanism  in  the  transmission  of 
force :  there  can  be  no  total  gain.  What  is  gained  in  force  is 
lost  in  movement,  and  in  many  complicated  machines  the 
theoretical  comparison  of  force  applied  and  resultant  force 
may  be  ascertained  by  the  measures  of  their  movements. 

The  resultant  effects  of  forces,  as  heretofore  treated,  have 
been  without  motion,  or  static.  But  when  motion  is  produced, 
the  forces  are  called  dynamic.  A  weight  suspended  or  sup- 
ported exerts  a  force,  which  is  balanced  by  the  resistance  of 
the  suspending  or  supporting  medium ;  but  a  falling  weight 
acquires  an  increasing  velocity  with  every  unit  of  time  or 

space  passed.  All  bodies  would  fall  with  the  same  velocities  were  it  not  for  the  different 
resistances  from  the  air  due  to  their  different  bulk  in  proportion  to  their  weight.  Dense 
articles,  as  stones  and  metals,  acquire  a  velocity  in  this  latitude  of  about  32'2  feet  in  each 
second,  called  the  intensity  of  gravity,  or  g.  The  value  of  g  at  the  equator  is  32'088 ;  at 
the  poles,  32-253.  A  body 


FIG.  377. 


Starting  with  a  velocity 

Falls  during  the  1st  second 

Acquiring  a  velocity  of 

Falls  during  the  2d  second  

Acquiring  a  velocity  of  twice  32,  or 

Falls  during  the  3d  second. 

Acquiring  a  velocity  of  3  X  32=  

Falls  during  the  4th  second 

Acquiring  a  velocity  of  4  X  32  = 

Falls  during  the  5th  second 

Acquiring  a  velocity  of  5  X  32  = 


\32  feet  per  second. 


32+  16\= 

\ 
\64  feet  per  second. 


Ft.     Tot.  Fall. 
16         16 


48        64 


32+32+16\=     80 

96  feet  per  second. 


32+ 


32+J32+  32+  16\=      112        256 

j \128  feet  per  second. 


32+32  + 


16\  =  .  144        400 


' J 160  feet  per  second. 


MECHANICS. 


211 


FIG.  378. 


Calling  s  the  space  passed  over,  «  the  terminal  velocity  in  feet,  t  the  time  in  seconds  of 
falling,  *  =  igt*,  v=gt  or  =  V64.4a.  In  determining  the  velocity  of  issuing  water  under  a 
head  A,  corresponding  to  s  in  the  equation,  it  is  generally  near  enough  to  reckon  «  as  eight 
times  the  square  root  of  the  head  (VA). 

The  motion  of  falling  bodies  is  a  uniformly  accelerated  one,  but  there  are  also  uni- 
formly retarded  motions  in  which  the  velocity  is  decreased  by  equal  losses  in  equal  times. 
There  are  also  uniform  motions  when  bodies  are  impelled 
by  a  constant  force  and  opposed  by  constant  resistances. 

In  Fig.  378,  o  s  represents  the  trace  of  a  body  impelled 
horizontally  by  a  uniform,  but  falling  through  the  action  of 
gravity  with  an  accelerated,  force.  This  curve,  a  parab- 
ola, represents  approximately  the  curve  of  the  thread  of 
stream  issuing  from  an  orifice,  or  flowing. 

It  will  be  seen  that  to  produce  twice  the  velocity  the 
body  must  fall  through  four  times  the  space;  that  there  is 
four  times  the  force  stored  in  the  body.  But  to  main- 
tain this  velocity  uniformly,  only  twice  the  force  is  neces- 
sary. The  momentum  of  a  body  is  its  mass  multiplied  by 
its  velocity,  but  its  inertia  is  as  the  square  of  the  velocity. 
It  is  an  established  principle  of  mechanics  that  the  results 
must  be  proportional  to  the  causes:  if  a  body  has  to  be 
raised  four  feet  to  obtain  a  double  velocity  in  falling,  the 
destructive  result  of  that  fall  must  also  be  four  times. 

Under  statics,  it  has  been    shown  that  forces  may  be 
resolved  and  compounded.     The  same  may  be  done  dynamically- — that  which  has  been 
treated  as  weight  must  now  be  considered  as  momentum. 

In  treating  of  dynamic  forces  the  resultants  have  been  considered  as  equal  to  the  exer- 
tion, without  any  losses  by  resistances.  This  never  happens  in  practice ;  the  resist- 
ances are  a  very  large  element.  Resistances  from  the  medium  in  which  the  bodies  are 
moved  are  from  the  surfaces  oh  which  the  bodies  are  supported  ;  resistances  due  to 
the  displacement  of  the  fluid  in  which  the  bodies  move,  and  fric- 
tional resistances,  or  what  is  termed  skin-resistances,  of  bodies 
moving  through  air  or  water;  and  the  surface-resistance  of  bod- 
ies sliding  or  rolling  on  each  other.  Suppose  a  weight  to  rest 
on  a  horizontal  surface — it  will  take  a  certain  force  to  move  the 
insistent  weight  depending  on  the  amount  of  this  weight  and 
the  kind  of  surfaces  in  contact,  and  the  force  that  will  just 
cause  motion  overcomes  the  friction,  or  frictional  force,  and  is 
equal  to  it.  The  frictional  force  is  only  a  percentage  of  the  in- 
sistent force  of  the  body,  and  this  percentage  is  called  the  co-efficient  of  friction.  If  the 
horizontal  surface  of  support  be  raised  at  one  end,  so  as  to  make  the  surface  inclined,  it 
will  after  a  time  become  so  steep  that  the  insistent  body  will 
slide  down  the  surface.  Thus,  in  Fig.  379,  if  the  body  Q  is 
ready  to  slip  on  the  surface  A  B,  the  angle  BAG,  which  rep- 
resents the  angle  of  the  surface  with  the  horizontal,  is  called 
the  angle  of  repose,  or  limiting  angle  of  frictional  resistance; 
or  thus  (Fig.  380),  if  the  force  acting  in  the  direction  P"  M 
is  just  sufficient  to  produce  motion  of  the  mass  M  along  the 
plane  F  Q,  the  angle  P  M  P"  is  the  limiting  angle  of  resist- 
ance. 

General  Morin  has  made  an  elaborate  course  of  experiments  on  friction,  the  results  of 
which  are  given  in  the  table  on  page  212.  It  was  formerly  held  that  friction  was  directly 


M 

FIG.  379. 


212 


MECHANICS. 


as  the  weight,  without  regard  to  the  amount  of  surface  or  velocity  of  movements.  And 
M.  Morin's  experiments,  as  rather  applicable  to  the  friction  of  quiescence  and  slow  move- 
ments, come  within  this  rule.  But  in  practice  it  has  been  found  that  the  co-efficient  of 
friction  with  unguents  is  reduced  by  increase  of  velocity  and  temperature ;  that  extent 
of  surface  maybe  prejudicial;  and  that  careful  selection  of  unguents,  according  to  the 
work  to  be  done,  must  be  made  to  economize  power  by  the  reduction  of  friction. 

Mr.  0.  I.  H.  Woodbery,  in  his  experiments  on  the  driving  of  cotton-spindles,  found  the 
co-efficient  of  friction  to  be  from  7  to  20  per  cent,  the  lond  being  from  one  to  five  pounds 
per  square  inch  ;  while  Professor  Thurston,  with  heavy  loads  of  1,000  pounds  per  square 
inch,  as  on  the  crank-pins  of  the  North  River  steamboat-engines,  found  the  co-efficient  of 
friction  was  one  half  of  one  per  cent,  the  unguent  being  sperm-oil.  Practically  it  may  be 
said  that  the  co-efficient  of  friction  for  light-running  spindles  should  not  exceed  10  per 
cent,  and  for  the  usual  work  in  shops,  of  say  100  to  200  pounds,  should  not  exceed  from 
2  to  3  per  cent. 

EXPERIMENTS   ON   FRICTION,  BY   M.    MORIN. 


SURFACES  OF  CONTACT. 

WITHOUT  UNGUENTS. 

UNCTUOUS  SURFACES. 

FRICTION  OF 
MOTION. 

FKICTION   OF 
QUIESCENCE. 

FRICTION  OF 
MOTION. 

FRICTION   OF 
QUIESCENCE. 

Co-efficient 
of  friction. 

Limiting 
angle  of 
resistance. 

Co-efficient 
of  friction. 

Limiting 
angle  of 
resistance. 

Co-efficient 
of  friction. 

Limiting 
angle  of 
resistance. 

Co-efficient 
of  friction. 

Limiting 
angle  of 
resistance. 

Oak  upon  oak,  fibers  parallel  to  the 
motion 

0-478 

0-324 
0-246 

25°  33' 

17-58 
13-50 

0-625 

0-540 
0-376 

32°   1' 

28-28 
20-87 

0-108 

0-143 
0-136 
0-140 

6°  10' 

8°  9' 
7-45 
7-59 

0890 
0-314 

21°  19' 
17°  26' 

Oak  upon  oak,  fibers  of  the  moving 
body,  perpendicular  to  the  motion.  .  . 
Oak  upon  elm,  fibers  parallel  

Wrought-iron  upon  oak                          | 

0-619 
0-133 
0-194 
0-172 
0-195 
0-152 

0-147 
0-217 
0-161 
0201 
0-296 

31°  47' 
7-52 
10°  59' 
9-46 
11-3 
8-39 

0-619 
0-137 
0-194 

0-i62 

31°  47' 
7-49 
10-59 

9-is 

"     wrought-iron  
41     cast-iron  
44               "     brass 

0-177 

o'-ieo 

0-125 
0-144 
0-143 
0-132 
0-107 

103 

9-6' 
7-8 
8-12 
8-9 
7-32 
6-7 

o'-iis 

6-44 

Cast-iron  on  elm  .... 

'"         "  cast-iron 

44         "  wrought-iron  
44         "  brass  

8-22 
12-15 
9-9 
11-22 
1630 

.... 

Brass  upon  cast-iron                                ! 

44        "     brass                  ...       .         - 

.... 

.... 

0-184 

7-88 

o-iei 

9-19 
14-57 

Leather  ox-hide,  well  tanned,  on  oak.. 
u        on  cast-iron,  wetted.  .  j 
belts  on  oaken  drums  ' 
41        cast-iron  pulleys  
Common  building  -  stones  upon  the 
same  

0-229 

12-54 

2-67 

0-27 
0-28 
(  0-38  to 
|0-65 

0-47 

20-49- 
882 

0-65— 
0-75 

38-2- 
36-53 

MECHANICAL   WORK    OK    EFFECT. 

Mechanical  work  is  the  effect  of  the  simple  action  of  a  force  upon  a  resistance  which 
is  directly  opposed  to  it,  and  which  it  continuously  destroys,  giving  motion  in  that  direc- 
tion to  the  point  of  application  of  the  resistance  ;  it  is,  therefore,  the  product  of  two  indis- 
pensable qualities  or  terms : 

First. — The  effort,  or  pressure  exerted. 

Second. — The  space  passed  through  in  a  given  time,  or  the  velocity. 

The  unit  of  force  in  England  and  here  is  represented  by  the  pound,  and  the  unit  of 
space  by  the  foot. 

The  amount  of  mechanical  work  increases  directly  as  the  increase  of  either  of  these 
terms,  and  in  the  proportion  compounded  of  the  two  when  both  increase.  If,  for  example, 
the  pressure  exerted  be  equal  to  4  pounds,  and  the  velocity  one  foot  per  second,  the  amount 
of  work  will  be  expressed  by  4x1  =  4.  If  the  velocity  be  double,  the  work  becomes 
4x2  =  8,  or  double  also ;  and  if,  with  the  velocity  double,  or  2  feet  per  second,  the  press- 
ure be  doubled  as  well — that  is,  raised  to  8  pounds — the  work  will  be  8x2  =  16  pounds 


MECHANICS.  213 

feet.  It  is  more  usual  to  write  foot-pounds,  but  we  invariably  use  the  former,  following 
the  Continental  idiom  of  kilogrammetre,  in  which  the  unit  of  force,  kilogramme,  precedes 
that  of  space,  the  metre. 

In  comparison  of  motors  with  each  other,  it  is  usual  to  speak  of  them  as  so  many  horse- 
power equivalent  to  550  pounds  feet  per  second,  or  33,000  pounds  feet  per  minute.  The 
Continental  horse-power  is  equal  to  75-  kilograinmetres  or  54:2*48  pounds  feet  per  second. 

It  is  very  common  to  use  other  units  of  force  and  space,  as  tons-miles ;  and  train-miles, 
in  railway  practice. 

The  time  must  also  be  expressed  or  understood.  It  is  impossible  to  express  or  state 
intelligibly  an  amount  of  mechanical  effect,  without  indicating  all  the  three  terms — force, 
space,  and  time. 

The  motors  generally  employed  in  manufactures  and  industrial  arts  are  of  two  kinds — 
living,  as  men  and  animals  ;  and  inanimate,  as  water  and  steam. 

What  may  be  termed  the  amount  of  a  day's  work,  producible  by  men  and  animals,  is 
the  product  of  the  force  exerted,  multiplied  into  the  distance  or  space  passed  over,  and  the 
time  during  which  the  action  is  sustained.  There  will,  however,  in  all  cases  be  a  certain 
proportion  of  effort,  in  relation  to  the  velocity  and  duration,  which  will  yield  the  largest 
possible  product  or  day's  work  for  any  one  individual,  and  this  product  may  be  termed  the 
maximum  effect.  In  other  words,  a  man  will  produce  a  greater  mechanical  effect  by  ex- 
erting a  certain  effort  at  a  certain  velocity,  than  he  will  by  exerting  a  greater  effort  at  a 
less  velocity,  or  a  less  effort  at  a  greater  velocity,  and  the  proportion  of  effort  and  velocity 
which  will  yield  the  maximum  effect  is  different  in  different  individuals. 

In  the  manner  and  means  in  which  the  strength  of  men  and  animals  is  applied,  there 
.are  three  circumstances  which  demand  attention : 

1.  The  power,  when  the  strength  of  the  animal  is  exerted  against  a  resistance  that  is 
at  rest. 

2.  The  power,  when  the  stationary  resistance  is  overcome,  and  the  animal  is  in  motion. 
And, 

3.  The  power,  when  the  animal  has  attained  the  highest  amount  of  its  speed. 

In  the  first  case,  the  animal  exerts  not  only  its  muscular  force  or  strength,  but  at  the 
same  time  a  very  considerable  portion  of  its  weight  or  gravity.  The  power,  therefore, 
from  these  causes  must  be  the  greatest  possible.  In  the  second  case,  some  portion  of  the 
power  of  the  animal  is  withdrawn  to  maintain  its  own  progressive  motion  ;  consequently, 
the  amount  of  useful  labor  varies  with  the  variations  of  speed.  In  the  third  case,  the 
power  of  the  animal  is  wholly  expended  in  maintaining  its  locomotion;  it  therefore  can 
carry  no  weight. 

Weisbach  calls  the  mean  effort  of  an  animal  one  fifth  its  weight,  which  may  serve  as  a 
general  rnle,  but,  in  practice,  will  be  considerably  modified,  when  applied  to  the  indi- 
vidual, depending  upon  the  exertions,  and  the  conditions  and  circumstances  under  which  it 
is  made.  A  man-power  is  usually  estimated  at  one  sixth  of  a  horse-power  (H.  P.)  ;  yet,  if 
the  muscular  force  of  a  man  be  required  for  an  effort  of  short  duration,  it  will  exceed  one 
liorse-power.  Thus,  n  horse-power  is  equal  to  33,000  pounds  feet  per  minute,  or  550 
pounds  feet  per  second;  and,  if  a  man  weighing  150  pounds  move  up-stairs  at  the  rate  of 
four  feet  per  second,  he  exerts  a  force  of  600  pounds  feet,  which  he  can  readily  double  for 
a  few  seconds. 

The  force  of  a  man  is  utilized  mechanically  through  levers,  as  in  pumping  or  rowing, 
or  at  a  vertical  capstan,  or  at  a  crank,  carrying  or  dragging  loads,  shoveling,  etc.  In 
continuous  work  at  the  lever  he  will  exert  from  25  to  30  pounds ;  at  the  crank,  from  15 
to  20  pounds.  • 

The  muscular  force  of  horses  is  utilized  in  the  draft  of  carriages,  in  hoisting,  and  in 
horse-powers,  either  moving  in  a  circle  round  a  central  shaft  or  on  a  revolving  platform, 
or  on  an  endless  belt.  The  draught  of  a  horse  varies  with  the  speed  of  movement  and  its 


214: 


MECHANICS. 


duration.  Trautwine  gives  the  draught  of  a  horse  at  two  and  a  half  miles  per  hour  for  10 
hours  per  day,  100  pounds;  8  hours,  125  pounds;  6  hours,  166f  pounds;  5  hours,  200 
pounds.  The  omnibus-horses  here  average  nearly  six  miles  per  hour,  and  make  16  to  24 
miles  per  day;  the  average  will  not  exceed  16  miles.  At  the  Manhattan  Gas  Works,  a 
span  of  horses  hoist  from  the  lighter  200  tons  gross  in  10  hours  to  the  height  of  say  25 
feet,  with  charges  of  6  to  the  ton,  in  a  bucket  weighing  150  pounds,  the  rope  passing  over 
a  single  block  and  through  a  snatch-block.  On  a  horse-power,  the  force  exerted  by  a  single 
horse  is  from  125  to  175  pounds,  at  an  average  speed  of  about  three  miles  per  hour,  and  for 
eight  hours  per  day.  Beyond  a  speed  of  four  miles  per  hour,  the  pounds  foot  of  work  of  a 
horse  will  decrease  in  an  increasing  ratio  up  to  the  limits  of  his  speed,  when  the  whole 
work  done  will  be  used  up  in  locomotion.  In  proportioning  levers,  cranks,  traces,  chains, 
through  which  animal  force  is  transmitted  to  machines,  or  for  mechanical  purposes,  it  ia 
not  safe  to  estimate  the  stress  as  the  average  force ;  there  are  impulses  and  stresses  in 
action  which  will  exceed  the  weight  of  the  animal. 

Water-Power. — Water,  used  for  the  purposes  of  power,  moves  machinery  either  by  its- 
weight,  by  pressure,  by  impact,  or  by  reaction,  and  is  applied  through  various  forms  of 
wheels.  However  used,  the  mechanical  effect  inherent  in  water  is  the  product  of  its 
weight  into  the  height  from  which  it  falls  ;  but  there  are  many  losses  incurred  in  its  appli- 
cation, so  that  only  a  portion  of  the  mechanical  effect  becomes  available ;  and  the  com- 
parative efficiency  of  any  water-wheel  or  motor  is  represented  by  this  percentage  of  the 
absolute  effect  of  the  water  applicable  to  power. 

The  quantity  of  water  supplied  to  the  mills  at  Lowell,  permanently,  for  the  working 
hours  per  day  is  about  4,000  cubic  feet  per  second,  and  the  entire  fall  33  feet.  In  the  dis- 
tribution of  the  water  by  the  canals  about  two  feet  of  fall  is  lost,  and  the  mill-powers,  as 
leased  to  the  mills,  would  be  about  4,000  cubic  feet  per  second,  on  a  31 -foot  fall.  In  the 
passage  of  the  water  through  the  trunks  or  pent-stocks  to  the  wheels,  and  from  the  wheels 
to  the  river  or  other  canals,  there  is  still  another  loss  of  head,  which  may  be  considered 
about  one  foot,  so  that  the  net  fall  is  only  30  feet. 

4,000  cu.  ft.  x  62-33  weight  of  water  per  cu.  ft.  x  30  ft.  fall 

550  Ibs.  ft.  per  H.  P.  per  sec. 
=  13,600  horse-power. 

But  only  a  percentage  of  this  power  is  available  for  mechanical  power.     The  efficiency 
of  the  turbines,  the  wheels  now  generally  in  use  here,  may  be  taken  at  80  per  cent  of  the 
gross  horse-power.     The  net  horse-power  will  then  be  13,600  x  '80  =  10*880  horse-power. 
Wind  is  applied  for  the  purposes  of  power ;  but,  as  there  is  no  constancy  in  its  action, 
its  use  is  mostly  confined  to  the  purpose  of  raising  water 
by  means  of  pumps  into  cisterns  or  reservoirs. 

Steam  is  the  elastic  fluid  into  which  water  is  converted 
by  a  continuous  application  of  heat.  It  is  used  to  pro- 
duce mechanical  action  almost  invariably  by  means  of  a 
piston  movable  in  a  cylinder.  Thus,  in  Fig.  381,  the  steam 
entering  through  the  lower  channel-way,  or  port,  presses 
against  the  under  side  of  the  piston  in  the  direction  of  the 
arrow,  the  piston  is  forced  upward,  the  steam  above  the 
piston  escaping  through  the  exhaust-channel  o.  When 
the  piston  reaches  the  top  of  the  cylinder,  the  valve  is 
changed  by  mechanism,  the  steam  enters  above  the  pis- 
ton, and  the  steam  below  it  escapes  through  the  exhaust. 
In  this  way  a  reciprocating  movement  is  established.  To 
determine  the  horse-power  of  a  steam-engine,  multiply  the  area  of  the  piston  in  square 
inches  by  the  effective  pressure  in  pounds  on  each  square  inch  of  piston,  and  the  product 
by  the  travel  in  feet  through  which  the  piston  moves  per  minute,  and  divide  this  last 


FIG.  381. 


MECHANICS. 


215 


product  by  33,000.    The  travel  is  the  length  of  stroke  multiplied  by  the  number  of  strokes, 
or  double  the  number  of  revolutions  per  minute. 

Example. — Let  the  diameter  of  the  piston  be  18  inches,  the  effective  pressure  45  pounds 
per  square  inch,  the  stroke  30",  the  revolutions  60,  or  300  feet  travel  per  minute,  what  will 
be  the  horse-power  of  the  engine  ? 

Area  of  piston,  254-46  square  inches. 
254-46  x  45  x  300 

-337000 =  104'  h<^-P°wer. 

As  steam  in  its  passage  through  channels  and  in  the  cylinder  is  subject  to  various 
losses  of  pressure,  and  as  the  steam  is  worked  under  more  or  less  expansion,  and  as  the 
exhaust  steam  is  discharged  under  more  or  less  pressure,  whether  into  the  air  or  into  a 
condenser,  it  is  impossible  to  determine  the  effective  pressure  except  by  the  means  of  an 
indicator. 

The  principle  of  working  steam  expansively  is  as  follows :  If  a  cubic  foot  of  air  of  the 
atmospheric  density  be  compressed  into  the  compass  of  half  a  cubic  foot,  its  elasticity  will 
be  increased  from  15  pounds  on  the  square  inch  to  30  pounds  ;  if  the  volume  be  enlarged 


100 


10 


to  two  cubic  feet,  the  pressure  will  be  one  half,  or  Y|  pounds.     The  same  law  holds  in  all 
other  proportions  for  gases  and  vapors,  provided  their  temperature  is  unchanged. 

Fig.  382  illustrates  this  graphically.    Suppose  the  piston  in  the  cylinder  to  have  made 
one  tenth  of  its  stroke,  and  to  be  at  .1,  and  the  pressure  at  100  pounds  above  the  absolute  0 


216 


MECHANICS. 


(or  vacuum)  to  which  expansion  is  referred,  and  not  to  the  atmospheric  line  representing 
nearly  15  pounds  pressure:  if  the  steam-valve  be  now  closed,  and  the  piston  he  moved  to 
the  position  .2,  the  space  occupied  by  the  steam  will  be  double  what  it  was  at  first,  and 
the  pressure  one  half,  J-f2-,  or  50  pounds.  If  the  piston  be  moved  to  .3,  the  pressure  will 
be  -J-,  or  33$-  pounds  ;  to  .4,  J,  or  25  pounds  ;  and  so  on  to  .5,  .6,  .7,  .8,  .9,  .1.0,  the  pressure 
will  be  £,  -J,  -f,  £,  |-,  TV ;  and,  at  the  end,  the  expansion  will  be  said  to  be  ten  times,  and 
the  cut-off  (or  where  the  steam  was  shut  off  from  the  cylinder),  at  ^  of  the  stroke. 

When  the  steam  is  cut  off,  if  there  be  no  leak  through  the  valves  or  by  the  piston,  this 
quantity  may  be  considered  constant,  although  there  are  losses  by  condensation  from  the 
surfaces  of  cylinder  and  piston,  and  the  conversion  of  heat  into  work.  But  it  will  generally 
be  found  that  the  weight  of  steam,  as  represented  by  the  volumes,  will  be  greater  at  the 

end  of  the  stroke  than  at  the  cut-off,  owing  to  re- 
evaporation  of  condensed  or  conveyed  water  by  the 
cylinder  surfaces. 

To  illustrate  the  theoretical  advantages  of  a  cut- 
off, draw  lines  across  the  card  (Fig.  382)  at  40  and  20 
pounds.  The  portions  of  the  card  below  these  lines 
will  represent  the  card  of  an  engine,  working  at  an 
initial  pressure  of  40  pounds,  and  cutting  off  at  .25, 
or  £  stroke.  The  portions  below  the  20-pound  line, 
the  card  of  an  engine,  with  this  initial  pressure  cut- 
ting off  at  .5,  or  £  stroke.  The  original  card  and  these 
other  cards  use  equal  quantities  of  steam,  but  the  work 

is  very  different;  in  the  first,  all  the  work  is  below  the  40-pound  line,  and  in  the  other 
all  below  20  pounds. 

Of  late  compound  steam-engines  have  become  very  popular.  They  consist  of  two  cyl- 
inders, a  high-pressure  (h.  p.  c.)and  a  low-pressure  (i.  p.  c.)  one.  Fig.  383  shows  the  gen- 
eral arrangement,  but  without  the  valves.  The  h.  p.  c.  (A,  B,  0,  D)  draws  its  steam  from 
the  boiler  and  exhausts  into  the  1.  p.  c.  (A',  B',  C',  D') ;  the  top  of  the  h.  p.  c.  into  the  bottom 


of  the  1.  p.  c.,  and  vice  versa,  so  that  the  pressure  on  the  pistons  of  the  two  cylinders  is  in 
the  same  direction. 

For  the  comparison  of  the  theoretical  effect  of  the  single  cylinder  and  compound  en- 
gines, construct  the  card  of  a  single  cylinder,  Fig.  384,  shown  in  dotted  line,  in  which  0'8  is 
the  length  of  stroke,  50  pounds  the  initial  pressure,  and  .2  the  point  of  cut-off.  If  50,  C,  .2,  .0 
represent  the  h.  p.  c.  of  a  compound,  and  the  cylinder  be  filled  at  the  pressure  of  50  pounds, 


MECHANICS. 


217 


the  quantity  of  steam  used  at  each  stroke  will  be  the  same  as  in  the  single  cylinder,  and, 
to  expand  equally  with  this,  the  stroke  of  the  1.  p.  c.  is  represented  by  .2,  .1,  .0.  When  the 
piston  of  the  h.  p.  c.  is  about  to  commence  its  stroke  downward,  for  instance,  the  cylinder 
beneath  it  is  full  of  steam  at  50  pounds.  As  the  steam  rushes  in  above  the  piston  from 
the  boiler,  the  steam  below  the  piston  begins  to  exhaust  into  the  upper  part  of  the  1.  p.  c., 
and  consequently  falls  off,  while  the  pressure  above  the  h.  p.  c.  piston  in  connection  with 
the  boiler  maintains  its  50  pounds. 

When  expansion  commences  in  the  1.  p.  c.,  the  pressure  is  the  same  as  in  the  h.  p.  c., 
50  pounds  ;  but  the  expansion  takes  place  differently  from  that  in  a  single  cylinder.  At  the 
end  of  the  first  eighth  of  the  stroke,  the  space  in  the  1.  p.  c.  is  equal  to  ^  that  of  the  h.  p.  c., 
but  its  space  has  been  reduced  by  the  movement  of  the  piston  £;  therefore,  the  space  now 
occupied  by  the  steam  is  £  +  •£ ,  or  ^  of  what  it  was  before  expansion,  and  the  50  pounds 

becomes  —  -  =  36-4  nearly.  At  £  stroke,  the  space  in  the  1.  p.  c.  is  equal  to  that  of  the 
h.  p.  c.,  and  in  the  h.  p.  c.  it  is  reduced  to  f;  the  total  space  is  now  1  +  £  =  £,  and  the 
expansion  is ,  or  28-6  pounds  nearly.  At  £  stroke  the  space  is  1|  x  £ ,  and  the 

pressure,  consequently,  23'5  nearly.  At  the  end  of  the  stroke  the  space  of  the  h.  p.  c.  is 
entirely  shut  off,  and  that  of  the  1.  p.  c.  filled  with  expanded  steam,  at  12£  pounds,  J  of 
the  initial  pressure.  The  full  line,  C  T',  represents  the  expansion  as  it  has  taken  place  in 
the  1.  p.  c. ;  but,  as  said  above,  the  pressure  below  the  piston  in  the  h.  p.  c.  falls  off  as 
expansion  goes  on  in  the  1.  p.  c.  The  pressure  in  the  1.  p.  c.  at  the  top  is  the  same  as  the 
h.  p.  c.  at  the  bottom,  and,  if  these  pressures  be  transferred  to  the  h.  p.  c.  card,  there  will  be 
a  curve,  50  T",  which  will  represent  the  back  pressure  beneath  the  h.  p.  c.  piston.  The  back 
pressure  is  shown  in  the  shaded  portion,  above  which  is  the  net  pressure  on  the  piston ;  if 
these  net  pressures  be  divided  by  4,  and  plotted,  as  shown,  above  the  1.  p.  c.  expansion, 
curve  C  T',  then  the  curve  C  H  will  represent  the  curves  of  pressures  of  the  united  h.  p.  c. 
and  1.  p.  c.  on  the  same  scale  as  that  of  the  single  cylinder. 

Figs.  385,  386  represent  these  cards,  both  on  the  same  scale,  and  it  will  be  observed 
that,  theoretically,  there  is  no  difference  in  effect  between  steam  used  in  a  single  cylin- 


FIG.  385. 


FIG. 


der  or  in  a  compound.  But,  practically,  the  compound  is,  for  many  purposes,  found  the 
most  economical,  due  in  part  to  the  smaller  condensation,  since  the  surfaces  in  the  h.  p.  c. 
are  never  cooled  below  the  limit  of  expansion,  in  example  12£  pounds  (204°),  while  the 
1.  p.  c.  and  the  single  cylinder  are  cooled  to  the  limit  of  condensation,  or  probably  about 
126°. 

In  addition,  comparing  the  two  cards  (Figs.  385,  386),  it  will  be  observed  that  the  forces 
in  the  compound  cylinders  are  less  irregular  than  in  the  single  cylinder,  and  the  necessi- 
ties of  a  fly-wheel,  to  equalize  forces  and  resistances,  are  less. 

The  cards  of  the  compound  engines  above  drawn  do  not  take  into  consideration  the 
loss  of  pressure  in  the  channels  between  the  h.  p.  c.  and  1.  p.  c.,  and  there  is  a  class 
of  compound  engines  in  which  the  h.  p.  c.  exhausts  into  an  intermediate  chamber,  be- 


218 


MECHANICS. 


tween  it  and  the  1.  p.  c.,  to  which  the  construction  of  cards  given  is  not  applicable.     They 
can  best  be  determined  from  practical  examples. 

The  above  illustrations  represent  purely  the  theoretical  card.  The  vacuum  is  perfect, 
and  the  steam  in  the  cylinders  at  full  pressure,  both  in  introduction  and  at  relief,  without 
any  wire-drawing,  reduction,  or  rounding,  incident  on  actual  practice. 

Fig.  387  represents  a  real  card  taken  from  a  steam-cylinder  of  a  condensing  engine. 
To  determine  the  mean  effective  pressure,  divide  the  atmospheric  line,  embraced  in  the 
card,  into  20  equal  parts,  and  draw  ordinates  through  the  .1,  .3,  .5,  .7,  .9,  .11,  .13,  .15,  .17, 

.19th  divisions.  The  lines  embraced  be- 
tween the  card  outlines  represent  the  pres- 
sure at  different  parts  of  the  stroke — .05, 
.15,  and  so  on,  on  the  scale  of  the  indicator- 
spring  ;  these,  added  together  and  divided 
by  10,  give  the  mean  effective  pressure  (in. 
e.  p.)  on  this  card,  43'4  pounds. 

The  mean  effective  pressure  multiplied 
by  the  area  of  piston,  in  square  inches,  by 
the  length  of  stroke,  and  number  of  strokes 
per  minute,  gives  the  pounds-feet  of  work 
per  minute,  which,  divided  by  33,000,  will 
give  the  indicated  horse-power  (i.  h.  p.)  of 
the  engine. 

To  determine  whether  a  steam-engine  is 
working  properly,  it  is  necessary  to  compare 
the  absolute  card  with  the  theoretical  one. 
Fig.  388  represents  an  indicator-card,  as 
taken  from  a  steam-cylinder  in  which  there 
is  no  condensation ;  the  exhaust  is  directly  into  the  air.  On  this  is  shown  the  construction 
of  the  isothermal  curve.  It  will  be  observed  that  there  is  a  line,  A  B,  to  the  top  of  the  card. 
The  space  between  this  and  the  card  represents  the  spaces  between  the  cylinder-head  and 
piston,  and  between  the  steam-valves  and  the  cylinder,  called  the  clearance,  which  are 
estimated  in  percentages  of  the  capacity  of  the  cylinder,  and  is  thus  plotted  on  the  indica- 
tor-card. On  the  indicator-card,  as  taken  by  the  instrument,  the  absolute  0  can  not  be 
taken,  but  only  that  of  the  atmosphere,  the  0  will  be  at  a  distance  below  this,  correspond- 
ing to  the  barometric  pressure,  usually  14'8  pounds.  Draw  the  0  line  parallel  to  the  at- 
mospheric line,  the  clearance  line  perpendicular  to  it,  a  line  parallel  to  the  0  line,  at  the 
height  of  the  initial  pressure,  and  a  line  parallel  to  the  clearance  line  at  the  point  of  cut- 


FIG.  387. 


FIG.  388. 


FIG.  389. 


off  on  the  initial  pressure  line.  Any  point  on  the  expansion  line,  as  la,  22,  32,  may  be  de- 
termined by  drawing  lines  B  1,  B  2,  B  3,  and  then  horizontal  lines  13  li,  2a  2i,  32  3i  from, 
their  intersections  li,  2i,  81.  With  the  cuj-off  line,  parallel  to  the  0  line,  and  perpen- 
diculars from  1,  2,  3,  the  intersections  of  these  two  lines,  12.  2a,  32,  will  be  the  points  in  the 


MECHANICS. 

curve.  The  curves  in  the  outline  of  the  cards,  at  the  times  of  admission,  cut-off,  and 
exhaust,  show  the  action  of  the  valves  and  time  occupied  in  change  of  condition.  The 
stroke  commences  at  A,  cuts  off  at  C,  commences  to  exhaust  at  E ;  about  D  the  exhaust- 
valve  closes,  and  the  steam  between  the  piston  and  the  ends  of  the  cylinder  begins  to  be 
compressed,  and  the  curve  developed  is  called  the  curve  of  compression. 

In  expanding,  steam  does  not  maintain  the  same  temperature  ;  there  is  a  fall  of  tem- 
perature, and  consequently  less  space  occupied  than  shown  by  the  isothermal  curve;  the 
curve  thus  developed  is  called  the  adiabatic  curve.  In  Fig.  389  the  construction  of  this 
curve,  the  line  Ce,  through  the  point  of  cut-off,  is  inclined  to  the  line  A  B,  1°  43',  to  which 
the  lines  1  la,  2  2a  are  drawn  parallel,  but  otherwise  the  same  as  in  the  preceding  figure ; 
practically,  the  isothermal  curve  corresponds  more  nearly  with  that  formed  by  the  cards, 
as,  especially  near  the  end  of  the  stroke,  there  is  considerable  transmission  of  heat  from 
the  cylinder  surfaces  to  the  steam,  more  than  that  lost  by  mere  expansion. 

The  indicator-cards  show  very  fairly  the  amount  of  power  exerted  on  the  piston,  but 
they  do  not  show  the  economy  of  the  whole  machine  including  the  boilers.  The  boilers 
may  be  faulty,  in  that  they  do  not  evaporate  sufficient  water  for  the  coal  consumed,  or  that 
the  ebullition  is  too  local  and  violent,  without  sufficient  steam-space,  so  that  water  is 
taken  off  with  the  steam ;  or  the  steam-cylinder  and  its  working  may  be  faulty,  in  that  the 
steam  is  condensed  therein  without  doing  any  work.  The  economic  value  of  the  boiler 
may  be  determined  by  the  measure  of  the  quantity  of  water  pumped  into  the  boilers,  and 
the  quality  of  the  steam. 


MACHINE    DESIGN    AND    MECHANICAL 
CONSTRUCTIONS. 

IN  the  designing  of  new  machines  and  mechanical  constructions,  the  draughtsman  must 
draw  from  his  knowledge  of  well-known  forms  and  parts,  and  combine  them;  but,  to  pro- 
portion them  properly,  and  adapt  them  to  the  purposes  required,  he  must  understand  the 
stresses  to  which  they  are  to  be  subjected,  and  the  action  and  endurance  of  the  material  to 
be  used,  to  withstand  these  stresses. 

In  the  present  technical  application  of  the  term,  stress  is  confined  to  a  force  exerted 
between  two  bodies  or  parts  of  a  body,  such  as  a  pull,  push,  or  twist.  Strain  is  the  altera- 
tion produced  by  a  stress.  Stress  is  the  cause,  strain  the  effect ;  the  first  is  measured 
by  the  load,  the  latter  by  the  deformation  of  the  body  produced  by  the  first.  A  stress,  not 
greater  than  the  elastic  limit  of  the  material  acted  upon,  produces  a  strain  which  disappears 
as  soon  as  the  load  is  removed :  up  to  this  limit  the  strain  is  proportional  to  the  stress ; 
beyond,  the  strain  increases  faster  than  the  stress,  up  to  the  point  of  rupture.  The  elastic 
limit  is  a  percentage  of  the  breaking  strain,  varying  with  the  kind  of  material  and  applica- 
tion of  stress.  Stress  is  usually  designated  as  load,  meaning  thereby  the  sum  of  all  the 
external  forces  acting  on  the  member  or  structure,  together  with  its  weight. 

Dead  load,  or  weight,  is  a  steady,  unchangeable  load.  Live  loads  are  variable,  alternately 
imposed  and  removed,  or  varying  in  intensity  or  direction.  It  is  usual,  in  designing  con- 
structions, to  proportion  the  parts  to  resist  a  much  greater  load  than  will  be  brought  on 
them  in  the  structure  ;  the  load  is  multiplied  by  a  factor  termed  factor  of  safety,  as  a  secu- 
rity against  imperfections  in  material  and  workmanship,  contingencies  of  settlement,  and 
other  incidental  stresses.  But  it  must  be  observed  that  these  imperfections  are  such  as 
can  not  be  seen  and  met ;  there  can  be  no  factor  of  safety  to  provide  for  poor  and  unknown 
material  and  defective  workmanship. 

The  factor  of  safety  adopted  for  dead  loads  varies  but  little  with  the  same  kind  of  ma- 
terial ;  but,  for  live  loads,  the  factor  varies  not  only  with  the  material,  but  with  the  char- 
acter of  the  stresses,  whether  they  are  applied  and  relieved  gradually  or  suddenly  ;  whether 
they  only  vary  in  intensity,  or  also  in  direction,  alternately  compressive  or  tensile.  In 
this  latter  case  the  load  should  never  be  considered  less  than  the  sum  of  the  stresses,  with 
a  large  factor  of  safety.  Vibrations,  shocks,  and  changes  in  the  direction  of  stresses,  con- 
centrate the  strains  at  the  weakest  point  of  the  construction,  and  rupture  takes  place  at 
these  points,  which  would  be  adequate  to  the  strain  if  the  form  throughout  were  uniform 
with  that  at  these  points.  Thus,  boiler-plates  show  wear  just  at  the  edge  of  the  lap  of 
the  sheets,  and  car-axles  (Fig.  390),  with  sharp  angles  at  the  journals,  are  known  to  break 
after  a  time,  while  under  the  same  stresses  an  axle  of  uniform  size  with  the  journal  would 
not  break;  nor  if  a  slight  curve  or  rim  £  inch  radius  (Fig.  391)  be  made  in  the  angle  to 
distribute  stress. 

Besides  provisions  for  strength,  the  draughtsman  should  understand  the  necessities  of 
the  construction,  and  the  character  of  the  material  to  be  used.  He  should  know  what 
parts  of  the  design  are  to  be  forged,  cast,  framed,  and  how  it  is  to  be  done.  He  should 


MACHINE   DESIGN  AND  MECHANICAL   CONSTRUCTIONS.  221 

know  what  wear  is  to  be  met,  and  what  waste,  as  rust  or  rot,  to  be  provided  for.  This 
knowledge  can  only  be  arrived  at  by  reference  to  examples  of  practice  and  by  observation 
of  results  under  similar  conditions  of  use  and  time. 

The  stresses  to  which  constructions  and  parts  of  constructions  are  subjected  are  the 
tensile  or  stretching  stress,  tending  to  lengthen  a  body  in  the  direction  of  the  stress ;  the 
compressive  or  crushing  stress,  tending  to  shorten  a  body  in  the  direction  of  the  stress; 
the  shearing  or  cutting  stress,  tending  to  elongate,  compress,  and  deflect;  the  torsional  or 


FIG.  390.  FIG.  391. 

twisting  stress,  the  effect  being  an  angular  deflection  of  the  parts  of  the  body ;  and  the 
transverse  or  lateral  stress,  tending  to  bend  the  body  or  break  it  across. 

At  page  195  is  given  a  table  of  the  strength  of  various  metals  to  resist  compression  and 
tensional  stresses,  and  examples  will  hereafter  be  given  of  varied  constructions,  with  their 
usual  or  required  factors  of  safety  ;  but,  for  a  practical  rule  for  the  common  necessities  of 
the  above  stresses,  under  dead  loads,  10,000  pounds  per  square  inch  for  wrought-iron  may 
be  considered  perfectly  safe. 

Posts  in  structures  are  subjected  to  compressive  stresses ;  but,  as  the  action  is  modified 
somewhat  by  a  tendency  to  bend,  depending  on  the  proportion  of  the  length  to  the  diame- 
ter, and  the  material  of  which  they  are  composed,  the  usual  tables  of  crushing  strength 
are  not  generally  applicable,  and  the  formulas  to  be  depended  on  are  those  deduced  from 
practical  tests.  The  best  tests  of  wooden  posts  are  those  made  by  Professor  Lanza,  for  the 
Boston  Manufacturers'  Mutual  Fire-insurance  Company,  and  the  following  are  the  results : 

"  That  the  strength  of  a  column  of  hard  pine  or  oak,  with  flat  ends,  the  load  being  uni- 
formly distributed  over  the  ends,  is  practically  independent  of  the  length,  such  columns 
giving  way  by  direct  crushing,  the  deflection,  if  any,  being  very  small.  Tests  were  on 
columns  6"  to'  10"  diameter  x  12  feet.  The  average  crushing  strength  of  very  highly- 
seasoned,  hard  pine  was  7,386  pounds  per  square  inch.  Some  very  slow-growth  and 
highly-seasoned,  9,339  pounds;  very  wet  and  green,  3,015  pounds;  seasoned  about  three 
months,  3,400  pounds;  not  very  well  seasoned  and  not  very  green,  4,400  to  4,700  pounds. 
The  average  of  two  specimens  of  thoroughly-seasoned  white-oak,  7,150  pounds;  for  green 
and  knotty,  average,  3,200  pounds.  Spruce,  nearly  5,000  pounds.  Whitewood,  3,000 
pounds. 

"That  it  is  a  mistake  to  turn  columns,  taper,  or  even  turn  them  at  all,  square  columns 
being  much  stronger,  cheaper,  and  better,  and  that  accuracy  of  fitting  is  of  great  conse- 
quence, that  the  stress  may  be  directly  vertical."  The  professor  recommends  that  longitu- 
dinal holes  be  bored  through  the  center  of  columns  to  allow  of  the  circulation  of  air  (in 
the  experiments  the  holes  were  I'l"  diameter),  and  that  iron  caps  be  used  instead  of  wooden 
bolsters,  as  the  wooden  bolster  will  fail  at  a  pressure  far  below  that  which  the  column  is 
capable  of  resisting,  and  the  unevenness  of  pressure  brought  about  by  the  bolster  is  some- 
times so  great  as  to  crack  the  column.  He  also  recommends  horizontal  holes  in  the  iron 
caps  to  connect  the  longitudinal  ones  in  the  column  with  the  outer  air. 

From  the  whole  of  the  experiments,  we  estimate  the  safe  load,  for  fair-grained,  well- 
seasoned  oak  or  yellow-pine  columns  to  be  from  1,000  to  1,500  pounds  per  square  inch ;  for 
the  more  imperfect  and  green  specimens,  from  300  to  500  pounds ;  for  good  specimens  of 
whitewood,  about  300  pounds ;  and  of  spruce,  about  500  pounds. 

Cast-Iron. — For  the  columns  of  buildings  where  the  load  is  dead,  cast-iron  is  very  gen- 
erally used.  They  are,  in  interiors,  mostly  of  circular  section,  but  for  outer  columns  forms 
are  used  suited  to  the  necessities  of  their  position  or  style  of  architecture.  They  admit  of 


222 


MACHINE   DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


considerable  ornamentation  and  finish  direct  from  the  mold ;  but,  as  they  are  liable  to  de- 
fects not  readily  detected  in  the  process  of  casting,  the  factor  of  safety  is  usually  taken  as 
high  as  5.  To  protect  them  against  the  effects  of  fire  and  water  in  conflagrations,  they 
are  often  covered  with  an  outer  shell  of  cast-iron  or  plaster,  or  of  both. 

The  experiments  of  Hodgkinson  are  the  usual  basis  of  all  formulae  on  the  strength  of 
circular  cast-iron  columns,  and  the  ends  of  all  columns  are  now  required  to  be  faced  by 
architects  and  by  the  rules  of  building  departments,  since  Mr.  Hodgkinson  states  this  rule, 
that  "  in  all  long  columns,  of  the  same  dimensions,  the  resistance  to  fracture  by  flexion  is 
three  times  greater  when  they  are  flat  and  firmly  bedded  than  when  they  are  rounded  and 
capable  of  moving." 

Table  of  the  safe  load  of  solid  cylindrical  columns,  with  flat  ends  calculated  with  a  fac- 
tor of  safety  of  5. 

TABLE  OF  SAFE  LOADS  FOR  SOLID  CAST-IRON  COLUMNS,  WITH  FLAT  ENDS. 


Diam. 

8' 
1,000 
Ibs. 

9' 
1,000 
Jbs. 

10' 
1,000 

Ibs. 

11' 
1,000 
Ibs. 

12' 
1,000 
Ibs. 

13' 
1,000 
Ibs. 

14' 
1,000 
Ibs. 

16' 

1,000 
Ibs. 

16' 
1,000 
Ibs. 

17' 
1,000 
Ibs. 

18' 
1,000 
Ibb. 

19' 
1,000 
Ibs. 

20' 
1,000 
Ibs. 

21' 
1,000 
Ibs. 

22' 
1,000 

lb». 

23' 
1,000 
Ibs. 

24' 
1,000 
Ibs. 

8" 

29- 

23-  j      20- 

IT- 

14- 

13- 

ii- 

10- 

9- 

8- 

7- 

T- 

6- 

6- 

5- 

5- 

4- 

3*" 

40- 

81- 

26-        22- 

19- 

17- 

is- 

13- 

12- 

If 

10- 

9- 

8- 

T- 

7- 

8- 

6- 

8?" 

SO- 

41- 

84-        29-        25- 

22- 

iy 

17-        15- 

14- 

12- 

ii- 

10- 

10- 

9-        S- 

8- 

3J" 

63- 

54" 

43-        37"        32- 

23- 

24- 

2-2-        19'       18- 

16- 

is- 

13- 

12- 

11-  !  11 

10- 

4" 

77- 

66-        54-        46    ;      40' 

35- 

31- 

27-        24-       22- 

20- 

18- 

17-      15- 

14-       18- 

12- 

44" 

li- 

80- i      70- 

57-  ;     49- 

4:3- 

38- 

34-        80-       27" 

25 

23- 

21 

19- 

18- 

16- 

15- 

4i" 

no- 

96-  j      84- 

74-        61' 

53- 

47"        41- 

87' 

S3" 

30- 

28- 

25- 

23- 

22' 

20- 

19- 

4*' 

130- 

113-        99- 

88-  !      73- 

64- 

56- 

50-        45- 

41- 

87- 

34 

31- 

28- 

ze- 

24- 

23- 

5 

152- 

183- 

117- 

103-  ]      92' 

77" 

68-        60' 

54' 

49- 

44- 

40- 

37- 

34- 

st  • 

29- 

27' 

54" 

1715- 

154-      136-  1    121- 

10S' 

97" 

81-        72- 

64- 

£8- 

53- 

48- 

44- 

40- 

37- 

85- 

32- 

4' 

201- 

177-      157-      140- 

125- 

113 

95' 

85- 

76- 

68' 

62" 

57' 

52- 

48' 

44- 

41- 

38- 

5J" 

230- 

203-      180-      161- 

144- 

13J- 

US'        99-  I      89- 

80- 

73- 

66' 

01- 

se- 

52- 

48- 

45- 

6  ' 

260- 

2W-      205-      183- 

165- 

149- 

135- 

115-      1(13- 

93' 

84' 

77' 

71- 

es- 

60- 

56' 

52- 

tij" 

2J2- 

260- 

232-  i    203- 

187' 

169- 

154'       140'       lilt- 

108- 

98- 

89- 

82" 

75- 

69- 

64' 

60- 

€*' 

827- 

292"      2H1-      234- 

212- 

192- 

174-  !    159-      146- 

124' 

112- 

102- 

94* 

86- 

80- 

74- 

09- 

3" 

304- 

326' 

292-  i    263-      238' 

216- 

197- 

180-       165- 

141- 

128' 

117- 

107- 

99- 

91- 

85- 

7'J- 

7' 

404- 

362- 

325-      293-1    266' 

242- 

221- 

202-  !    186- 

ni- 

146' 

133- 

122- 

112- 

104- 

96- 

90- 

74' 

445- 

400- 

861- 

326- 

296- 

269- 

246-      226- 

208- 

192- 

177- 

151- 

138-    127' 

IIS-     109' 

101- 

If' 

489- 

441- 

3)8- 

861- 

328- 

299- 

274-      251- 

231-     214- 

198- 

170-    15(5-     148- 

183-     123-    114' 

"•*' 

536- 

4S4- 

438- 

398- 

862- 

331- 

303- 

278' 

257'     '2:;7- 

220' 

294'  1  175- 

Iftl- 

149-    138-     128- 

8'' 

584- 

529- 

480- 

436- 

398- 

364- 

334- 

80S' 

284-     263- 

244" 

227- 

i  '.)»;•    iso- 

167-    155- 

144' 

^i  ' 

689- 

626- 

571- 

521- 

477' 

437- 

402' 

871- 

343- 

818- 

296- 

275- 

257-    241- 

2(17'    192-    178- 

9  ' 

802- 

733- 

670- 

614- 

564- 

519- 

479 

442' 

410- 

881- 

3^4-    331- 

309-    290- 

272-    235-  '  218- 

«*'' 

926- 

849- 

780- 

717- 

660- 

609' 

563- 

522- 

484-     451- 

420-    393- 

867-    34.V 

824-    305-    265- 

10  ' 

1058- 

975- 

898- 

829- 

765- 

708' 

650- 

61)9- 

566-     528- 

493'  |4G1- 

432- 

406- 

382-  :36U-    840- 

10*" 

1195- 

1108- 

1026' 

957' 

892- 

848- 

779-      740- 

693-     658- 

610-    580- 

546-    511- 

485-    459'  1433- 

11" 

1359- 

1264- 

1159- 

1083- 

1017- 

950' 

889'      846' 

793-     7.r)l- 

703- 

665' 

627- 

589- 

561-    542-    513- 

11*" 

1517- 

1413- 

1319- 

1226- 

1147- 

1080- 

1018-       956- 

904  ' 

852- 

810-    758- 

727- 

691- 

655-    (518-    587- 

12'' 

1674- 

1583-  !  1470- 

1880- 

1289-    1221- 

1142-     1074'     1018- 

973- 

916- 

871-     746"     701- 

667-    645- 

Gil' 

1                                                          i                                i 

Solid  columns  are  very  seldom  used  in  constructions;  they  are  almost  invariably  made 
hollow,  the  shell  being  i"  to  2"  thick.  To  determine  the  safe  load  of  a  hollow  column,  it 
will  be  sufficiently  accurate  to  take  from  the  table  the  safe  load  of  a  column  equal  to  that 
of  the  exterior  diameter,  and  subtract  from  this  the  safe  load  of  a  column  of  a  diameter 
equal  to  the  core. 

Example. — To  find  the  safe  load  of  a  column  12  feet  long,  8"  exterior  diameter,  shell  £". 

Safe  load  of  8"  column 398,000  Ibs. 

"       "      u  6V      "      212,000    " 

"       "    required  column 186,000    " 

For  square  box-columns,  it  will  be  safe  to  estimate  that  a  square  column  will  support  as 
much  as  a  round  one,  the  side  of  the  one  being  equal  to  the  diameter  of  the  other,  and  the 
thickness  of  shell  the  same. 

For  a  star-column  (Fig.  392),  the  load  should  be  about  £  less  than  on  a  cylindrical  col- 
umn of  same  diameter  and  same  area  of  section. 

Wrought-Iron  Columns. — With  the  decrease  in  the  cost  of  the  manufacture  of  shapes  in 
wrought-iron,  columns  of  this  material  have  largely  superseded  those  of  cast-iron  in  con- 


MACHINE  DESIGN  AND  MECHANIC 

structions  liable  to  varying  loads  and  shocks.     Fig. 
column,  Fig.  394  of  the  Piper,  Fig.  395  of  the  Keystone 
The  Phoenix  columns  vary  in  the  number  of  segments, 


223 


ws  the  section  of  a  Phoenix 


FIG.  393.  FIG.  394. 

TABLE  OF  PHOENIX  COLUMNS. 


FIG.  395. 


MAEK   OF   COLUMN. 

Thickness  in 
inches. 

Area  in  square 
inches. 

Weight  in  pounds 
per  foot. 

Internal  diam- 
eter. 

A  

, 

2'8 

9'3 

4  segments  

A 

5'8 

19'4 

*| 

B  

A 

5'0 

16-7 

4  segments. 

Ab 

14'8 

51 

*tf 

A 

5-8 

19'4 

4  segments  

f 

17' 

58'6 

6tt 

C  

A 

8'8 

30-3 

4  segments.  .  .  . 

IS 

40* 

138 

*A 

D  

1 

14-0 

48'2 

5  segments.  . 

4 

26- 

89'7 

H 

B..?  :"::'.: 

i 

16' 

55'2 

6  segments.  .  .  . 

il 

60' 

207' 

11 

F  

4 

24'5 

84'5 

7  segments  

A 

36'4 

125'6 

13 

G  

A. 

24- 

82'8 

8  segments  

IX 

80- 

276- 

Uf 

TABLE  OF  PIPER  AND  KEYSTONE  COLUMNS. 


4-iNcu  COLUMN. 

6-iNCH  COLUMN. 

8-iNCH  COLUMN. 

10-INCH 

COLUMN. 

Piper. 

Keystone. 

Piper. 

Keystone. 

Piper. 

Keystone. 

Piper. 

Keystone. 

Area,    Weight 

Area, 

Wight     Area, 

Weight    Area, 

Wight 

Area, 

Weight     Area, 

Wight 

Area, 

Weight 

Area, 

Weight 

sq.  in.     per  ft. 

sq.  in. 

per  ft. 

sq.  in. 

per  ft.     sq.  in. 

per  ft. 

14.  to. 

per  ft. 

sq.  in. 

per  ft. 

sq.  in. 

per  ft. 

sq.  in. 

per  ft. 

A 

5-2 

17-4 

5-6 

18-7 

i 

6'       20- 

7-3 

24-3 

7-1 

23-8 

11- 

36-6 

9'8 

32-6 

T^g- 

6-8     22-7 

8-4 

28-1 

8-7 

28-9 

12-5 

41-7 

11-8 

39-3 

16" 

53-3 

14-2 

47-4 

|^ 

7-6     25-3 

7-1 

23-7 

9-0 

31-8     10-2 

34- 

14- 

46-8 

13-8 

46' 

17-9 

59-7 

16-6 

55-3 

A 

8-4  ;  28' 

8-2 

27-3 

10-7 

35-6     11-7 

39-1 

15-6 

51'8 

15-8 

52-8 

19-8 

66- 

18-9 

63-1 

i 

9'3 

30-9 

11-8 

39-4     13-3 

44-2 

17-1 

56-9 

17-9 

59-5 

21-7 

72-3 

23-7 

78-9 

A 

14-8 

49-3 

18-6 

62- 

19-9 

66-2  j  23-6 

78-7 

26- 

86-7 

f 

16-3 

54-4 

20-1 

67-1 

21-9 

72-9    25-5 

85- 

28-4 

94-6 

H 

23-9 

79-6 

27-4 

91-3 

30-7 

102-4 

25-9 

86-4 

29-3 

97-7 

33-1 

110-3 

if 

• 

35-5 

118-2 

Figs.  396-399  are  sections  of  box-columns;  the  covers  of  398  and  399  must  be  made  in 
short  pieces,  to  admit  of  the  inside  riveting,  and  with  close  butt- joints  to  preserve  the 
strength.  The  thickness  of  the  webs  should  exceed  ^  of  the  width,  to  prevent  buckling 
under  stress. 


224  MACHINE   DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


™ 

FIG.  396. 


FIG.  397. 


FIG.  398. 


FIG.  399. 


.  /-\       r^       r\. 


IF 


FIG.  401. 


FIG.  402. 


FIG.  403. 


FIG.  404. 


FIG.  405. 


FIG.  406. 


FIG.  407. 


FIG.  408. 


FIG.  400. 


FIG.  409. 


FIG.  410. 


FIG.  411. 


Fig.  400  shows  the  elevation  and  section  of  an  open  or  lattice  column,  common  in  bridge 
and  railway  work.  In  estimating  strength  by  area  of  section,  in  lattice- columns,  the  areas 
of  continuous  support,  as  of  the  channel-irons,  a  5  and  c  d  in  the  figure,  are  only  considered. 

Figs.  401-403  are  sections  of  other  open  columns. 

Figs.  404-411  are  sections  of  various  forms  of  made-up  columns. 

The  caps  and  bases  are  usually  of  cast-iron  and  molded  to  the  requirements  of  po- 
sition. 

On  the  Strength  of  WrougJit-Iron  Columns. — The  upper  curve,  Fig.  412,  represents 
graphically  the  average  breaking  load,  taken  from  experiments  on  the  Phoenix,  Keystone, 
Piper,  and  open  columns,  with  flat  ends.  Horizontal  distances  give  the  proportions  of 

lengths  of  columns  to  diameters,  or  — — - — ,  the  vertical  distances  the  loads  in  pounds. 

diameter' 

The  lower  curves  represent  the  safe  loads,  under  factors  of  safety  of  3,  4,  and  5.  In  look- 
ing at  these  curves,  it  will  be  observed  that,  within  the  common  limits  of  practice,  of  15 

to  35  -^-     - — ,  these  lines  may  be  considered  straight ;    that  with  iron  of  a  breaking 

strength  of  52,000  pounds  per  square  inch,  and  within  the  above  limits,  and  a  factor  of 
safety  of  3,  the  safe  load  may  be  taken  at  11,000  per  square  inch ;  with  a  factor  of  safety 
of  4,  at  8,000  pounds ;  with  a  factor  of  safety  of  5,  at  6,500  pounds ;  and  that  for  common 
and  usual  purposes  10,000  pounds  per  square  inch  is  a  safe  load. 

It  has  generally  been  considered  that  columns  with  pin  or  cylindrical  ends  had  about 
f  of  the  resisting  strength  of  flat  ends,  but  if  the  pin-ends  are  closely  fitted,  so  that  the 
strains  are  uniformly  in  the  direction  of  the  length  of  the  column,  the  difference  is  but 
little  between  the  two  kinds  of  ends. 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


225 


The  sectional  areas  of  I,  channel,  and  angle  irons,  of  which  the  above  posts  are  com- 
posed, will  be  given  hereafter. 


50' 


Shearing  Stresses. — Parts  of  machines  and  of  constructions  subjected  to  these  stresses 
have  often  the  resistances  modified  by  friction,  combined  with  other  stresses.  The  sizes 
of  parts  necessary  to  resist  such  stresses  practically,  as  in  the  cases  of  bolts,  rivets,  and  the 
like,  will  be  hereafter  illustrated  by  examples  and  determined  by  particular  rules.  In 
general,  the  strength  to  resist  shearing  stress  is,  in  wrought-iron  and  steel,  from  70  to  80 
per  cent  of  its  tensile  strength ;  in  cast-iron,  about  40  per  cent  of  its  crushing  strength. 
The  softer  woods,  as  spruce,  white  pine,  hemlock,  resting  on  walls  or  girders,  will  safely 
sustain  a  load  of  200  to  300  pounds  per  square  inch  of  bearing  surface,  and  the  harder 
woods,  as  oak  and  Southern  pine,  300  to  500  pounds.  By  experiment,  oak  treenails,  1"  to 
If"  diameter,  were  found  to  have  an  ultimate  shearing  strength  of  about  two  tons  per 
square  inch  of  section ;  but,  according  to  Rankine,  the  planks  thus  connected  together 
should  have  a  thickness  of  at  least  three  times  the  diameter  of  the  treenails.  In  3"  planks, 
If"  treenails  bore  only  1'43  tons  per  square  inch  of  section;  in  6"  plank,  l'T3  tons. 

Torsional  Stress.— Every  shaft  through  which  power  is  transmitted,  whether  through 
gears,  cranks,  or  pulleys,  is  subjected  to  a  torsional  stress,  of  which  the  power  acting  tan- 
gentially  to  the  shaft  in  one  direction  is  resisted  by  the  load  in  an  opposite  direction. 
When  this  stress  exceeds  a  certain  limit  depending  on  the  material,  the  fibers  are  twisted 
asunder,  but  ranch  below  this  limit  the  elasticity  of  the  shaft  may  be  too  great  to  transmit 
power  uniformly. 

The  length  of  the  axle  subjected  to  torsion  does  not  affect  the  actual  amount  of  press- 
ure required  to  produce  rupture,  but  only  the  angle  of  torsion  which  precedes  rupture, 
and  therefore  the  space  through  which  the  pressure  must  be  made  to  act. 

A  torsional  deflection  of  1°  in  a  length  equal  to  twenty  diameters  of  the  shaft,  is  a  good 
working  limit  of  deflection— that  is,  -yfa  part  of  a  full  turn.  D.  V.  Clark  gives  the  follow- 
ing rule:  "To  find  the  diameter  of  a  shaft  capable  of  transmitting  a  given  torsional  stress 
within  good  working  limits.  Divide  the  torsional  stress  in  foot-pounds  by  18'5  for  cast- 
15 


226 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


iron;  27'7  for  wro light-iron ;  and  57'2  for  steel.  The  cube  root  of  the  quotient  is  the 
diameter  of  the  shaft  in  inches. 

Example. — On  the  teeth  of  a  4^-foot  gear,  the  force  exerted  is  2,800  pounds.  What 
should  be  the  diameter  of  a  wrought-iron  shaft  to  transmit  this  force  safely? 

The  torsional  stress  will  be  2,800  pounds  multiplied  by  the  radius  of  leverage,  2J  feet, 

6,300 
or  6,300  foot-pounds  =  -?—  ==  228,    ^228  =  6-1. 

27*7 

Transverse  Stress. — If  a  beam  supported  at  its  extremities  be  loaded  with  a  weight,  W, 
Fig.  413,  the  beam  is  subjected  to  a  bending  movement,  or  transverse  stress,  composed  of  a 
tensile  stress  on  the  lower  part  of  the  beam  and  compressive  stress  on  the  upper  part,  as 
will  be  readily  seen  by  the  figure.  In  addition,  the  weight  of  the  beam  and  its  load,  sup- 
ported on  the  abutments,  act  at  these  points  as  shearing  stresses. 


FIG.  413. 


FIG.  414. 


The  strength  of  a  square  or  rectangular  beam  to  resist  transverse  stress  is  as  the 
breadth  and  the  square  of  the  depth ;  and  inversely  as  the  length,  or  the  distance  from  or 
between  the  points  of  support.  Thus  a  beam  twice  the  breadth  of  another,  other  propor- 
tions being  alike,  has  twice  the  strength ;  or  twice  the  depth,  four  times  the  strength  ;  but 
twice  the  length,  only  half  the  strength. 

It  is  evident,  therefore,  that,  with  the  same  area  of  section,  the  deeper  a  beam  the 
stronger  it  will  be,  if  the  breadth  is  sufficient  to  prevent  lateral  buckling. 

To  cut  the  best  beam  from  a  log,  Fig.  414,  the  section  of  which  is  a  circle :  draw  a  diam- 
eter, divide  it  into  three  equal  parts,  erect  perpendiculars  at  the  points  of  division  1,  2, 
and  they  will  intersect  the  circumference  at  the  corners  of  the  beam,  of  which  the  ex- 
tremities of  the  diameter  are  the  other  two. 

Q  •*     j  2 

For  the  transverse  strength  of  rectangular  beams  the  general  formula  is  W  =  — - — ,  in 

which  W  is  the  breaking  weight ;  S,  a  number  determined  by  experiment  on  different 
materials;  5,  the  breadth,  and  d,  the  depth  in  inches;  and  Z,  the  length  in  feet. 

Figs.  415  to  422  represent  the  usual  methods  of  loading  beams,  and  the  loads  as  drawn 
represent  the  comparative  strength  of  beams  under  these  different  conditions.  Thus,  in 


FIG.  415. 


V////A 


FIG.  416. 


Fig.  415,  the  beam  supports  but  one  unit  of  load,  while  Fig.  416  supports  twice  as  much. 
The  formulae  given  represent  the  safe  dead  loads  with  a  factor  of  safety  of  6,  deduced  from 
experiments  of  Mr.  C.  J.  H.  Woodbury  on  Southern  pine.  For  spruce  the  co-efficient 
would  be  about  \  less,  and  for  live  loads  the  factor  of  safety  should  be  12. 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


227 


Beams  fixed  at  one  end  and  loaded  at  the  other  (Fig.  415). 
Id* 


Safe  load  =  30 


I 


Beams  fixed  at  one  end  and  load  distributed  uniformly,  not  as  represented  in  the 
figure,  as  the  two  units  of  weight  would  be  spread  over  the  whole  length  of  the  beam 

(Fig.  416). 

o  cL 
Safe  load  =  60  — . 

Beams  supported  at  the  extremities  and  loaded  at  the  middle  (Fig.  417). 

Id* 
Safe  load  =  120  — . 

• 


FIG.  417. 


FIG.  418. 


Beams  supported  at  the  extremities  and  the  load  uniformly  distributed  (Fig.  418). 

Id* 
Safe  load  =  240  -— . 

Beams,  one  end  firmly  fixed,  the  other  supported,  and  loaded  at  the  middle  (Fig.  419). 

•t      -j  g 

Safe  load  =  160-—. 
I 


^5^ 

^^ 

5Vr/'///V 

''/fty 

^^ 

'/"' 

yy>'yfl 

s^\^\ 

;| 

1 

/xx^ 

^~ 

FIG.  419. 


FIG.  420. 


Beams  with  one  end  fixed,  the  other  supported,  and  load  uniformly  distributed  (Fig.  420). 

Id* 

Safe  load  =  240  ——. 
I 

This  formula,  although  given  by  good  authorities,  is  evidently  too  small ;  it  should  be 

Id* 
probably  about  300  — — . 


FIG.  421, 


FIG.  422. 


228 


MACHINE  DESIGN  AND   MECHANICAL  CONSTRUCTIONS. 


Beams  with  both  ends  fixed,  and  loaded  at  center  (Fig.  421). 
Safe  load  240  =  —  —  . 

L 

Beams  with  both  ends  fixed,  and  load  uniformly  distributed  (Fig.  422). 
Safe  load  =  360  -^. 

t 

If  the  load  on  the  beam  be  neither  at  its  center  nor  distributed  as  in  Fig. 

423,  lay  off  on  any  convenient  scale  an  inclined  line,  A  C,  between  the  abut- 

ments, equal  to  the  weight  of  the 
load.  Let  fall  a  perpendicular  from 
the  bearing-point  of  the  load  to  this 
line  ;  it  will  divide  it  inversely  pro- 
portional to  the  load  on  the  abut- 
ments. In  the  figure,  the  load  is 
1,200  pounds  ;  the  perpendicular  in- 
tersects the  scale-line  beneath  at  900  ; 
900  pounds  is  therefore  the  load  on 
the  abutment  at  B,  and  the  balance 
of  the  weight,  or  300  pounds,  on  the 
abutment  A.  To  determine  the  size 
of  beam  of  uniform  section  to  resist 
the  bending  movements  of  the  loads, 

multiply  the  loads  on  the  abutments  together,  and  divide  by  one  quarter  of  the 

sum  of  the  two  loads.     Thus,  in  the  figure, 


Fm.  423. 


W'"' 


=  =  900>  the 


load  at  the  center  of  the  beam,  and  the  size  can  be  readily  determined  by  the 
formula  or  tables  given. 

If  the  load  is  not  distributed  symmetrically,  Fig.  424,  the  bending  move- 
ment and  shearing  stresses  may  be  readily  determined  graphically.  Let  loads 
equal  to  100,  365,  850,  and  125  pounds  be  supported  as  shown  by  the  beam  A  B 
(say,  12  feet).  At  one  side,  on  a  line  a  b,  perpendicular  to  the  beam,  lay  off  on 
any  convenient  scale,  100,  365,  850,  125,  to  represent  the  loads  on  the  beam  ;  from 
1,  2,  3,  4,  5  draw  lines  meeting  at  some  point,  C.  The  point  C  can  be  chosen 
anywhere,  but,  for  reasons  that  will  be  hereafter  self-evident,  it  will  be  better  to 
take  C  at  a  horizontal  distance  C  D  of  either  10,  100,  1,000,  etc.,  measured  on  the 
same  scale  as  the  loads  on  the  line  a  1).  From  the  points  of  support  of  the 
loads  on  the  beam  A  B,  let  fall  perpendiculars  ;  from  any  point  C,  on  line  A  C/ 
draw  the  line  C,  I,  parallel  to  C  1,  1,  2,  parallel  to  C  2,  2,  3,  parallel  to  C  3, 
3,  4/  parallel  to  C  4,  and  4,  F,  parallel  to  0  5.  Connect  C,  and  F,  and  draw 
the  line  C  F  parallel  to  this.  The  distance  1  F,  measured  on  the  scale  of  loads, 
will  give  the  reaction  in  pounds  on  the  abutment  equal  to  530,  and  5  F  =  910 
pounds  will  be  the  reaction  on  the  other  abutment  B.  These  are  shearing 
stresses,  and  their  sum  in  every  case  should  equal  the  sum  of  the  loads  —  in  this 
case,  1,440  pounds.  The  point  of  greatest  stress  in  the  beam  will  be  imme- 
diately above  the  longest  ordinate  in  the  polygon  C/  1,  —  F,  C,.  In  this  case 
it  will  be  at  the  point  of  support  of  the  850  pounds,  3,  3,,  being  the  longest  or- 
dinate in  the  polygon.  This  ordinate,  2*7,  measured  on  the  scale  of  the  beam 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


229 


FIG.  424. 


multiplied  by  the  horizontal  ordinate  C  D  (taken  here  at  1,000),  will  give  2,700. 
This  number,  divided  by  3,  one  quarter  of  the  span  A  B  of  the  beam,  will  give 
the  center  load,  equal  to  900,  for  which  the  size  can  be  determined  by  the 
formula  or  tables  as  before. 

TABLE  OF  THE  SAFE  CENTRAL  LOAD  OF  YELLOW-PINE  BEAMS,   CALCULATED 


FEOM  THE  FORMULA  —120 


bd* 

I  ' 


Span  in 
feet. 


4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 


DEPTH    IN    INCHES    OF   YELLOW-PINE    BEAMS,   ONE    INCH    WIDE. 


Sins. 

•tins. 

5  ins. 

GillS. 

7  ins.  8  ins. 

9  ins. 

10  ins. 

11  ins. 

12  ins. 

13  ins. 

14  ins. 

15  ins. 

16  ins. 
7680- 

270- 

480- 

750-  1080- 

1470'  1920- 

2430" 

3000' 

3630' 

4320'  5070- 

5880- 

6750- 

216- 
180- 
154- 
135- 
120- 

384- 

600- 

500- 

864- 
720- 
616- 

1176' 
980- 
840' 

735- 

1636- 
1280- 
1097- 
960' 
853' 

1944' 
1620- 
1389" 
1215' 

1080- 

2400- 
2000- 
1714- 
1500- 
1333- 

2904- 
2420- 
2074' 
1815- 
1613. 

3456- 

2880' 
2469' 
2160- 
1920' 

4056" 
3380' 
2897' 
2535- 
2253- 

4704- 
3920' 
3360' 
2940- 
2613' 

5400' 
4500- 
3857- 
3375- 
3000- 

6144- 
5120- 
4388- 
3840- 
3413- 

320- 
274- 
240- 
213- 

430- 
375- 
333' 

540- 
480- 

653' 

108- 

192' 
175- 

BOO- 
STS- 

432- 
392- 

588' 
535- 

768- 
700- 

972- 

1200- 
1092- 

1452. 
1320- 

1728- 
1571- 

2028' 
1844. 

2352- 
2140- 

2700- 
2457- 

3072- 
2793- 

882- 

160- 

250- 

360- 

490' 

640- 

810- 

1000- 

1210- 

1440' 

1690" 

I960- 

2250- 

2560- 

230- 
21S- 

332- 
308- 

452" 

420" 

592- 

648" 

747- 
693" 

923- 
860- 

1117- 

1328-  1560- 
1234-  1448' 

1808- 
1680- 

2070- 
1928- 

2363- 
2192- 

1037' 

288" 

392- 

512- 

648- 

800- 

968- 

1155'  [1352- 

1568- 

1800- 

2048' 

270- 
254- 

368- 
346' 

480- 
452- 

607' 
566" 

748- 
704- 

907' 

854- 

1080'  1267' 
1016"  1193- 

1470' 

1688- 

1588- 

1920' 
1808- 

1384- 

327- 

427* 

540- 

668' 

806- 

960-  1126' 

1307-  1500- 

1707' 

404' 
384- 

512' 
486" 

632- 

600' 

764- 

726- 

909' 
864- 

1067" 
1014- 

1238- 
1176- 

1422- 
1350- 

1616- 
1536- 

463- 

572- 

691- 

823- 

966- 

1120- 

1287' 

1463- 

442- 

546' 

660' 

785'  922- 

1070- 

1228- 

1395- 

522- 

631- 

752-  882- 

1023' 

1178- 

1329- 

600- 

605' 

720'  845- 

980' 

1125- 

1280- 

581- 

691-  811- 

940- 

1080- 

1230' 

558' 

665-  780' 

904- 

1035'  1182- 

230 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


This  table  is  deduced  from  Mr.  Woodbury's  experiments  on  yellow  pine, 
of  good  quality  and  practical  sizes.  For  spruce  he  takes  loads  of  about  one 
fifth  less. 

The  table  is  intended  to  be  used  as  a  unit  by  which  the  strength  of  timber  of 
usual  depths  and  spans  can  be  estimated,  by  multiplying  by  such  widths  as  are 
found  in  practice  ;  widths  of  less  than  two  inches  are  not  used.  The  strength 
given  in  the  table  is  in  excess  of  the  stiffness,  and  in  permanent  constructions 
it  is  necessary  to  proportion  the  beam  to  bear  its  load  with  a  certain  limited 
deflection.  Mr.  Woodbury  established  this  limit  in  wooden  beams  at  three 

quarters  of  an  inch  for  25-feet  span,  and  his  formula  is  E  =          ]      y  in  which 

0  / 1*  CL 

E,  the  modulus  of  elasticity  per  square  inch  is  for  Southern  pine  2,000,000, 
and  for  spruce  1,200,000  :  W  central  load  in  pounds,  I  the  span  in  feet,  I  the 
breadth,  h  the  depth,  and  d  the  deflection  of  beam,  all  in  inches.  Using  this 
formula,  we  have  drawn  marks  in  each  column  of  depth,  above  which  the 
loads  will  be  supported  stiffly,  and  below  less  so  than  recommended. 

It  is  to  be  observed  that  the  formula  is  applicable  to  seasoned  wood. 

Wooden  and  wrought-iron  beams  are  of  uniform  section  for  their  entire 
span,  but  cast-iron  can  be  readily  adapted  in  form  to  the  load  to  be  sus- 
tained. 

The  forms  of  beams  which  afford  equal  strength  throughout  are  parabolic 
(Figs.  425,  426,  427),  of  which  the  axis  A  B  and  the  vertex  A  are  given,  and 


A 


© 


FIG.  425. 


FIG.  426 


the  points  M  determined  by  calculations.     Figs.  426,  427  are  oftener  used 
when  the  force  is  applied  on  alternate  sides  of  A  B. 

A  beam  subjected  to  a  transverse  stress,  as  shown  in  Fig.  413,  one  side  is 
compressed,  while  the  other  side  is  extended  ;  and  therefore,  where  extension 
terminates  and  compression  begins,  there  is  a  lamina  or  surface,  g  h,  which 
is  neither  extended  nor  compressed,  called  the  neutral  surface.  As  the 
strains  are  proportional  to  the  distance  from  this  surface,  the  material  of 
which  the  beam  is  composed  should  be  concentrated  as  much  as  possible  at 
the  outer  surfaces,  as  can  readily  be  done  in  beams  of  cast  and  wrought  iron. 
Acting  on  these  principles,  Mr.  Hodgkinson  has  determined  the  most  econom- 
ical form  for  cast-iron  beams  or  girders,  of  which  the  section  is  given  (Fig. 
428);  it  has  been  found  that  the  strength  of  cast-iron  to  resist  compression  is 
about  six  times  that  to  resist  extension  ;  the  top  web  is  therefore  made  only 
one  sixth  the  area  of  the  lower  one.  The  depth  of  the  beam  is  generally  about 
one  sixteenth  of  its  length,  the  deeper  of  course  the  stronger  ;  the  thickness  of 
the  stem  or  the  upright  part  should  be  from  -J  an  inch  to  1£  inch,  according 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


231 


to  the  size  of  the  beam.  The  rule  for  finding  the  ultimate  strength  of  beams 
of  the  above  section  is :  Multiply  the  sectional  area  of  the  bottom  flange  in 
square  inches  by  the  depth  of  the  beam  in  inches,  and  divide  the  product  by 
the  distance  between  the  supports  in  feet,  and  2  '4:2  times  the  quotient  will  be 
the  breaking  weight  in  tons  (2,000  pounds).  As  has  already  been  shown 


FIG.  428. 

above,  the  section  thus  determined  need  only  be  that  of  the  greatest  strain, 
and  can  be  reduced  toward  the  points  of  support,  either  by  reducing  the 
width  of  the  flanges  to  a  parabolic  form  (Fig.  428),  or  by  reducing  the  thick- 
ness of  the  bottom  flange  ;  the  reduction  of  the  girder  in  depth  is  not  in 
general  as  economical  or  convenient. 

For  railway  structures  subject  to  an  impulsive  force,  Mr.  Joseph  Cubitt, 
C.  E.,  recommends  that  the  section  of  the  upper  flange  should  be  one  third 
that  of  the  lower. 

Fig.  429  is  side  elevation,  plan,  and  section  of  cast-iron  girder,  adopted  by 


FIG.  429. 


him  for  railway  purposes,  a  pair  of  girders  for  each  track,  the  rails  being 
supported  on  wooden  cross-beams. 


DIMENSIONS  FOR  DIFFERENT  SPANS. 


Opening. 

Bearing  on  ' 
abutment 

Height  of  girder 
at  center. 

Top  flange. 

Bottom  flange 
at  center. 

At  end. 

Thickness  of 
middle  web. 

12ft. 
30ft. 

l'-6" 

2'-6" 

l'-3" 
3'- 

3"  x  H" 
5"  x  2" 

l'-4"  x  If 
l'-6"  x  2" 

l'-8"     X  1|" 
l'-10"  x  2" 

H" 

2" 

45ft. 

2'-9" 

3'-9" 

7"  x  2f 

2''       x  2|" 

2''         x  2f 

2" 

232  MACHINE  DESIGN   AND  MECHANICAL  CONSTRUCTIONS. 

Some  years  since  the  bow-string  girder  was  in  very  common  use  in  this  city 
for  span  openings  of  from  fifteen  to  twenty-five  feet  in  the  fronts  and  rears  of 
stores  and  warehouses.  The  bow  was  made  of  cast-iron,  in  a  x-fora1?  and 
the  strings,  or  tension-rods,  were  of  wrought-iron.  In  this  composite  struct- 
ure it  was  impossible  to  calculate  the  strength  of  the  girder,  to  decide  how 
much  was  borne  by  the  bow  and  how  much  by  the  string.  The  strings  were 
forged  with  heads,  and  it  was  intended  that  the  fit  should  be  an  easy  one, 
so  that  some  compression  should  be  put  on  the  bow  before  tension  should  be 
put  on  the  rods.  But,  with  the  diminished  cost  of  wrought- 
iron,  cast-iron  girders  have  given  way  to  rolled  beams  and  box- 
girders  of  wrought  iron. 

Rolled  or  I  beams,  Fig.  430,  may  be  taken  as  the  type. 
They  are  made  at  many  rolling-mills.  The  depths  of  the 
beams  and  the  widths,  B?  of  bottom  and  top  flanges  do  not 
vary  much  with  the  different  makers  for  the  same  class  of 
beams ;  the  thickness  of  the  stems  varies  somewhat  more  pro- 
portionally. For  each  depth  there  are  usually  two  weights — 
the  light  and  heavy — and  are  thus  classed  in  the  trade,  as  light 
twelves  and  heavy  twelves,  and  lighter  or  heavier  weights  may 
be  made  to  order. 

There  is  considerable  difference  in  the  strengths  of  these  beams  as  given  in 
the  tables  of  the  different  makers :  in  the  table  on  page  233  we  have  tried  to 
modify  these  discrepancies  as  far  as  possible,  adopting  that  of  no  single  maker ; 
and  to  give  dimensions  such  as  will  suffice  for  the  purpose  of  the  draughtsman 
in  illustration,  with  tables  of  strength  which  can  be  relied  on  as  practical. 
We  have  discarded  the  usual  practice  of  stating  strength  in  tons,  and  have 
taken  100  pounds  instead,  so  that  00  need  only  be  added  to  the  tabulated 
figures  to  give  the  safe  distributed  load  in  pounds. 

It  is  assumed  in  these  tables  that  proper  provision  is  made  for  preventing 
the  beam  from  deflecting  sideways.  They  should  be  held  in  position  at  dis- 
tances not  exceeding  twenty  times  the  width  of  the  flange,  but  this  is  usually 
effected  by  the  necessities  of  the  construction,  the  brick  arches  between  the 
beams,  or  the  wooden  joists  resting  on  them.  The  beams  will  support  the 
loads  as  given  in  the  tables,  but  the  deflection  may  be  too  much  for  the 
purposes  to  be  served.  A  line  is  drawn  in  each  column  in  the  tables,  at 
which  the  deflection  is  -j-J-g-,  or  one  inch  for  every  thirty  feet  of  span,  beyond 
which,  if  the  beams  carry  plastered  ceilings,  the  deflection  is  apt  to  crack  the 
plastering. 

A  common  formula  for  determining  the  strength  of  a  wrought-iron  beam 

SV(a  +  ^)S 

is  W  =  -  — ,  in  which  W  is  the  load  in  pounds,  equally  distributed 

L 

on  the  beam,  D  the  effective  depth  between  the  centers  of  gravity  of  the  flanges, 
and  L  the  clear  span,  both  in  the  same  unit,  feet  or  inches  ;  a  the  area  of  the 
top  or  bottom  flange  in  square  inches  ;  a'  the  area  of  the  stem. 

To  find  the  sectional  area  of  a  beam-plate  or  rod  from  its  weight,  divide 
the  weight  per  yard  by  10 ;  and,  conversely,  to  determine  the  weight  per 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS.  233 


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234 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


linear  foot  from  the  sectional  area,  multiply  the  area  by  10  and  divide  the 
product  by  3. 

Thus,  if  a  bar  a  yard  long  weigh  40  pounds,  its  sectional  area  will  be  4 

Q     \/     1  A 

square  inches :  and  a  bar  of  9  square  inches  section  will  weigh  -  -  =  30 
pounds  per  foot. 

For  naval  constructions,  deck-beams,  Fig.  431,  are  rolled  at  different  mills, 
from  3"  to  12"  deep,  and  varied  widths  of  flanges  and  thicknesses  of  stem  ;  in 
general,  not  quite  up  to  the  grades  of  heavy  and  light  I-beams  in  weight,  but 
they  can  be  rolled  to  order  to  any  desirable  dimensions  within  the  limits  of 
depth  given.  Properly  proportioned,  they  should  be  equal  in  strength  to  the 
I-beams. 

Coupled  I-Beams. — When  the  load  is  beyond  the  strength  of  a  single  I- 
beam,  two  or  more  may  be  united,  as  shown  in  Fig.  432.  A  cast-iron  block,  or 


FIG.  432. 


FIG.  433. 


FIG.  434. 


FIG.  431. 


FIG.  435. 


FIG.  436. 


FIG.  437. 


FIG.  438., 


FIG.  439. 


FIG.  440. 


FIG.  441. 


FIG.  442. 


FIG.  443. 


FIG.  444. 


separator,  is  inserted  between  the  beams,  and  two  bolts,  passing  through  them 
and  the  block,  add  lateral  strength.  The  bolt-holes,  if  placed  at  some  distance 
from  the  center  of  the  span,  do  not  reduce  the  transverse  strength. 

It  is  not  unusual  to  strengthen  an  I-beam  by  the  riveting  of  a  plate  on  top 
(Fig.  433).  It  adds  to  the  areas  of  the  flanges  by  the  area  of  the  plate,  less 
that  of  the  rivet-holes  in  both  plate  and  flange. 

Box-girders  are  sometimes  made  up  in  the  same  way  by  two  Fs  and  plates 
across  top  and  bottom  (Fig.  434) ;  but,  as  the  access  to  the  inside  for  holding 
the  rivets  is  usually  impossible,  channel-beams  (Fig.  435)  are  preferred  for 
these  forms,  within  the  limits  to  which  these  beams  are  rolled. 

Channel-beams  can  be  furnished  of  depths  the  same  as  I-beams,  from  three 
to  fifteen  inches,  of  varied  grades  of  light  and  heavy,  and  within  any  desirable 
limits  of  weight. 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


235 


TABLE  OF  DIMENSIONS  OF  CHANNEL-BEAMS  IN  INCHES. 


DEPTH. 

Web. 
Thickness. 

FLANGE. 

Width. 

Thickness. 

3 

•2  to  '3 

1-51  to  1-61 

i  to  fer 

4 

•24  u  "39 

1-74 

1-89 

±"i 

5 

•25  "  '55 

1-93 

2'23 

6 

•23 

•53 

1-98 

2-28 

A  "  f 

7 

•30 

•55 

2-30 

2-55 

A  "  f 

8 

•30 

•75 

2-30 

2-75 

f  "it 

9 

•31 

•71 

2-43 

2-83 

1  "  * 

10 

•31 

•76 

2-56 

'  3-01 

f  "If 

12 

•46 

•96 

2-71 

'  3-21 

ft  "  l 

15 

•53 

•93 

3-53 

'  3-93 

i  "i 

It  may  be  desirable,  on  account  of  position,  to  finish  a  box-girder  as  in  Fig. 
436  ;  in  this  case  the  dimensions  must  be  such  as  to  admit  of  a  helper  inside 
to  hold  the  rivets.  Fig.  437  shows  a  closed  box-beam  made  of  channel-bars 
and  plates.  The  lower  channel  is  first  riveted,  and  the  upper  one  afterward. 
This  form  gives  a  clean  surface  below,  but  the  lower  channel-bar  can  be  re- 
versed and  riveted  the  same  as  the  upper. 

Where  the  purpose  can  be  served  by  I-beams,  either  single,  or  coupled, 
as  in  Fig.  432,  or  in  numbers,  they  afford  the  best  and  cheapest  construction. 
But,  where  the  spans  are  large  and  loads  heavy,  it  is  often  economical  to  obtain 
greater  depth  by  means  of  plate-girders,  as  in  Figs.  438,  439,  440,  441,  or  per- 
haps from  requirements  of  position,  as  in  Fig.  442,  subject  as  above  to  the 
necessities  of  large  inside  dimensions.  These  girders  are  made  up  of  plates 
of  uniform  thickness,  and  angle-irons  riveted  together. 

Angle-irons  are  made  of  varied  dimensions,  and  are  classed  as  equal-legged 
(Fig.  443),  unequal-legged  (Fig.  444),  and  square-root  angles  when  the  thick- 
ness of  the  iron  is  uniform  throughout,  and  consequently  the  interior  angle  a 
complete  right  angle  without  rounding.  The  following  table  gives  the  dimen- 
sions and  weights  of  the  angles  to  be  found  at  different  mills,  but  weights  can 
be  increased  to  order  : 

ANGLE-IRON.— WEIGHT  IN   POUNDS  PEE  FOOT. 


SIZE,  INCHES. 


AVERAGE  THICKNESS. 


t" 

A" 

±" 

•&" 

t" 

A" 

*" 

A" 

f" 

B" 

i" 

H" 

1" 

EQUAL  LEGS. 
6  x  6  

19'2 

21-7 

24-2 

26-7 

29-2 

31-7 

34-95 

4x4                  

9-5 

11'2 

12'9 

14-5 

16-2 

17-9 

19'5 

3£  x  3£           

8-3 

9-7 

11-2 

12'7 

14'1 

15'6 

17-0 

34  x  34%  . 

7-7 

9-0 

10-4 

11-7 

13'1 

14-4 

15'8 

3x3 

5-9 

7-2 

8-4 

9'7 

10'9 

12-2 

2£  x  2£ 

5-4 

6-5 

7'7 

8'8 

24  x  2£  

4'9 

5-9 

7-0 

8-0 

24*  x  24- 

3-5 

4'5 

5'4 

6'4 

7'3 

2x2  

3-1 

4-0 

4-8 

5-6 

1&  x  1&.  . 

2-1 

2-8 

3-5 

4-3 

5'0 

14  x  14 

1-8 

2-4 

3-0 

3-6 

14  x  14.. 

1-0 

1-5 

2-0 

14  x  14 

0-9 

1-4 

1'8 

1x1          

0'8 

1-2 

1-6 

&  x  £  ..  , 

0'6 

0-9 

236 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


ANGLE-IKON.— WEIGHT  IN  POUNDS   PEE  FOOT. 


SIZE,  INCHES. 

iVERAC 

E  THI 

CKNE88. 

1" 

A" 

*" 

&" 

1" 

A" 

i" 

A" 

1" 

tt" 

i" 

W 

r 

UNEQUAL  LEGS. 
6£  x  4  

14-6 

16-6 

18'6 

20-6 

22-7 

24-7 

26-7 

28'7 

6x4 

13-9 

16'0 

18'1 

20-2 

22-3 

24-4 

26  4 

28'4 

6  x  3£ 

11-3 

13'2 

15'0 

16-6 

18-4 

20-2 

22-1 

5  x  4  

10-8 

12'7 

14-5 

16'4 

18-3 

20-2 

22'0 

5x3^.. 

10-2 

11'9 

13-7 

15'5 

17-2 

19'0 

20-8 

5x3..                

9'5 

11-2 

19.-9 

14-5 

16'2 

17'9 

19'5 

4  x  3-J  

8-9 

10-5 

i?,-o 

13'6 

15-2 

16'7 

18'3 

4x3.. 

8'3 

9'7 

11-2 

12'7 

14-1 

15-6 

17'0 

3^  x  3  

7-7 

9'0 

10'4 

11-7 

13'1 

14-4 

15-8 

3i  x  2 

4-2 

5-3 

6-4 

7'4 

8-5 

3x2^  

4'4 

5-5 

6-7 

7-8 

9-0 

3  x  2  

4-0 

5-0 

6-0 

7*1 

8'1 

2£  x  2                    ... 

3-5 

4-5 

5-4 

6'4 

7'3 

21  x  14k  . 

2-5 

3-0 

3'8 

4-5 

2  x  If  

2-0 

2-6 

3-3 

4-0 

T-irons  (Fig.  445)  may  be  used  for  top  and  bottom  flanges 
in  the  manufacture  of  plate-girders,  by  riveting  a  web  on  one 
side  of  the  T,  or  on  both  sides,  with  a  separator  between  of  the 
thickness  of  the  stem  E  ;  but,  as  the  areas  of  section  of  T-irons 
to  be  had  are  small,,  the  flanges  will  be  too  slight  in  propor- 
tion to  the  webs  at  depths  above  that  of  rolled  beams.     Angle- 
irons  are  then  to  be  preferred  for  flanges.    The  T-irons  are  well 
adapted  in  many  positions  as  struts  or  braces,  and  can  be  bought  of  varied 
dimensions  and  weights,  from  widths,  B,  of  from  2  to  5  inches,  and  equal  or 
less  depths,  A,  and  thicknesses  from  -f$"  to  f ". 

Rivets  for  plate-girders  are  usually  from  £"  to  £"  diameter,  and  pitched  or 
spaced  not  more  than  6"  nor  less  than  3"  between  centers.  The  number  of  rivets 
through  flange  and  stem  are  the  same,  but  alternating.  Usually  angle  irons  and 
plates  can  be  had  of  the  full  length  of  girder,  but,  where  joints  are  necessary, 
they  should  be  butt,  with  a  splicing-piece  to  make  the  strength  as  nearly  as  pos- 
sible uniform.  Stiffeners  are  often  necessary  for  the  webs,  which  may  be  of 
band,  angle,  or  T  iron,  and  one  should  always  be  placed  at  each  end,  where  the 


shearing  stress  is  the  greatest. 

To  construct  a  diagram  from  the  formula,  W  = 


8  D  (a  +     )  S 


in    which 


the  relation  of  the  factors  may  be  shown.     Let  S  be  10,000,  on  account  of  loss 

of  strength  by  rivet-holes,  then  W  =  —  X  (a  +  -)  80,000.    On  a  sheet  of  cross- 

L  6 

section  paper,  from  a  corner,  0,  lay  off  on  the  line  of  ordinates,  5,  10,  15,  20, 

25,  representing  the  factor  a  +  -.     From  the  same  0,  on  the  line  of  abscissas, 

D  D 

i»  iV>  -h,  ih>>  ih,  A>   A,   iV>  representing  — .     Suppose  --  =  •&,  thenW  = 
/  ,  _L  Lt 

(a+  -)  2,000.     If  a  +  -  be  =  10,  then  W  =  20,000.    From  the  intersection  of 
6  6 

ordinate  on  line  of  3^,  and  abscissa  line  of  10,  draw  a  line  to  the  point  0.    This 


MACHINE  DESIGN   AND  MECHANICAL   CONSTRUCTIONS. 


237 


line  will  represent  the  safe  distributed  load  W,  and  its  intersections  of  the  or- 
dinates  and  abscissas  will  represent  the  relative  proportions  of  the  two  factors 

-  and  a  -\ —  under  this  load.     On  the  abscissa  line  15,  and  ordinate  fa,  AY 
L  6 

=  30,000,  on  line  20,  40,000,  and  so  on,  and  lines  drawn  from  these  intersec- 
tions to  0  will  represent  W. 

Fig.  446  is  thus  constructed,  but  lines  below  5  and  above  30  on  line  of  ordi- 
nates  are  erased,  as  within  these  limits  may  be  found  most  of  the  proportions 
required  in  practice. 


25 


20 


v 

/s 


y 

/IS 


20  /25 

FIG.  446. 


30 


"We  should  recommend  to  every  draughtsman  who  needed  this  sort  of  table 
to  construct  one  for  himself  on  cross-section  paper.  , 

Application  of  the  Diagram.  —  What  will  be  the  area  of  section  a+  --  of  a 
girder,  40-foot  span,  depth  32",  distributed  load  90,000  pounds? 

D  in  the  formula  represents  the  distance  between  the  centers  of  gravity  of 
the  flanges,  which  will  be  somewhat  less  than  the  depth  of  beam.  Approxi- 


mately  we  assume  it  at  30",  —  =  -  —  =  -jV,  and  tne  intersection  of  the  line 

L        30"  , 

of  load,  90,000,  with  the  ordinate  -fa,  will  be  18,  on  the  line  of  a  +  -.     A  fair 

6 

proportion  of  a  to  a'  is  5  to  6,  therefore  —  +  -  =  18  or  a'  =  18.     —  =  0.  6"  = 

66  30 

thickness  of  web,  and  a  =  f  of  18  =  15,  or  weight  per  foot  of  one  flange 

--  =  50  pounds,  which  is  slightly  in  excess  of  the  weight  of  two  angle- 
3 

irons  6  X  4  X  f,  compensated  by  thickness  of  web  outside  centers  of  gravity. 


238 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


This  calculation  is  sufficiently  near  for  all  practical  purposes,  but  D  can  be 
found  more  accurately  by  plotting  the  angle-irons  as  above,  on  thick  card- 
board, cutting  out  and  then  balancing  for  the  center  of  gravity. 

Composite  Beams. — Often,  in  constructions  where  the  beams  or  girders  are 
of  wood,  and  on  account  of  extent  of  spans  and  loads,  the  stress  is  beyond  the 
strength  and  stiffness  of  beams  of  this  material,  of  readily  available  dimen- 
sions, it  is  usual  to  supplement  by  some  application  of  iron.  A  simple  form,  in 
which  the  iron  is  not  exposed  to  view,  is  by  bolting  a  plate  or  flitch  of  wrought- 
iron  between  two  beams,  of  the  full  length  and  depth  of  the  beams,  and  of 
such  thickness  as  may  be  necessary.  In  bolting  them  together,  let  the  bolt- 
holes  be  so  bored  that  the  weight  of  the  beam  may  primarily  be  on  the  wood  ; 
the  stress  will  then  be  better  adjusted  between  the  two  materials  when  in  ser- 
vice. It  is  usual  to  make  the  holes  zigzag,  in  two  lines  about  one  quarter  the 
depth  of  beam  from  each  edge,  the  holes  closer  together  nearer  the  ends.  The 
safe-distributed  load  for  the  iron  may  be  estimated  from  the  formula  :  W.  = 

— ,  b  breadth,  h  depth,  I  length — all  in  inches. 

Fig.  447  represents  a  bracing  truss  of  wrought-iron  between  two  beams, 
which  should  be  let  into  the  wood.  As  it  is  held  firmly  laterally,  the  factor  of 


FIG.  447. 

safety  may  be  considered  about  one  third  of  the  crushing  resistance  of  the  ma- 
terial. The  load  on  each  inclined  bar  will  be  one  half  the  load  on  the  center, 
multiplied  by  the  length  of  the  bar  and  divided  by  the  rise.  Instead  of  wrought- 
iron,  cast-iron  or  wood  is  used. 

In  Fig.  448  the  beams  are  strengthened  by  a  tension-rod,  of  which  the 
strength  may  be  determined  by  that  of  the  material ;  allowing  the  usual  factor 


FIG.  448. 

of  safety,  the  load  is  obtained  as  in  the  example  above.  The  deeper  the  block 
beneath  the  center  of  the  beam,  the  less  the  stress  on  the  rods  for  the  same 
load.  In  construction,  the  beam  should  not  be  cambered  by  the  screwing  up 
of  the  rod  ;  but,  if  the  beams  are  crowning,  the  convex  side  should  be  placed  up- 
ward, the  nut  turned  by  hand  just  to  a  bearing,  and  the  tension  put  on  by  the 
settlement  of  the  beams  under  the  load. 

Fig.  449  represents  the  trussing  of  a  beam  by  two  struts  and  a  tension-rod. 
The  stress  on  the  tension-rod  is  the  load  on  c,  multiplied  by  the  length  a  d, 
divided  by  c  d. 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


239 


FIG.  449. 


The  theory  of  trusses  will  be  treated  and  illustrated  under  "Bridges"  and 
"Roofs,"  and  the  proportions  of  rivets  and  forms  of  plate-iron  joints  under 
"Boiler  Construction." 

Bolts  and  nuts  are  of  such  universal  application  that  their  manufacture 
forms  the  center  of  large  industries.  Much  thought  has  been  given  to  their 


FIG.  450. 


proportions  and  the  forms  of  thread,  but  without  producing  complete  uni- 
formity in  the  practice  of  different  countries  and  makers.  The  old  form  of 
thread  was  the  A  or  sharp  pitch  (Fig.  451),  still  used  by  some,  especially  when 
the  threads  are  cut  in  a  lathe.  In  this  country  the  standard  U.  S.  thread  is 


FIG.  451. 


that  recommended  by  the  Franklin  Institute  in  1864  (Fig.  452).  The  angle  is 
60°,  with  straight  sides  and  flat  surface  at  top  and  bottom,  equal  to  one  eighth 
the  pitch,  or  distance  from  center  to  center  of  threads. 


In  England,  the  standard  thread  for  bolts  and  nuts  is  the  Whitworth  (Fig. 
453) ;  the  angle  is  55°,  with  top  and  bottom  rounded. 


Z.oo 


FIG.  455. 


The  square  and  rounded  threads  (Figs.  454  and  455)  are  only  made  to  order 
and  used  in  presses  and  the  like  as  parts  of  machines. 


240 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


Figs.  456,  457,  and  458  represent  the  proportions  of  the  various  parts  of 
English  nuts  to  the  diameters  of  bolts,  as  1,  or  unity.  Fig.  457  is  a  flange- 
nut,  in  which  a  washer-like  flange  is  forged  with  the  nut. 


R 


1 


FIG.  456. 


FIG.  457. 


FIG.  458. 


Fig.  459  is  a  cap-nut,  in  which  the  thread  does  not  go  through  the  nut,  to 
prevent  leaking  along  the  thread,  and  a  soft  copper  washer  is  introduced  to  pre- 
vent leakage  below  the  nut. 

Figs.  460  and  461  are  circular  nuts,  in  one  of  which  holes  are  drilled  to 
insert  a  rod  for  turning,  and  in  the  other  grooves  for  a  spanner. 


FIG.  459. 


FIG.  460. 


FIG.  461. 


FIG.  462. 


Lock-nuts  (Fig.  462)  are  intended  to  prevent  the  gradual  unscrewing  of 
nuts  subjected  to  vibration,  which  is  to  a  great  extent  prevented  by  the  use  of 
double  nuts,  the  lock-nut  being  one  half  the  thickness  of  the  common  nut. 
The  usual  practice  is  as  shown,  the  lock-nut  being  outside  ;  the  better  way  is 
inside. 

The  following  figures  are  from  trade  circulars  ;  the  limits  of  sizes  given  are 
such  as  can  usually  be  found  in  stock. 

Figs.  463,  464,  and  465  are  machine-bolts,  from  i"  to  f"  diameter,  and  1" 


FIG.  463. 


to  4"  long,  but  not  flanged,  as  in  Fig.  463,  unless  expressly  ordered  -,  the  dot- 
ted line  shows  the  radius  of  curvature  of  a  finished  head.     The  diagonal  lines 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


241 


beneath  the  head  (Figs.  464  and  465)  represent  square  bolts  tapering  into  round 
bolts,  as  shown  by  the  curved  lines. 


Fio.  464. 


FIG.  465. 


Figs.  466  and  467  are  tap-bolts  and  set  screws,  from  i"  to  f  "  diameter,  and 
from  1"  to  3"  long. 


FIG.  466. 


FIG.  467. 


Fig.  468  is  a  carriage-bolt,  from  %'  to  f "  diameter,  and  from  1"  to  16"  long. 
Fig.  469  is  a  plow-bolt,  from  f"  to  £"  diameter,  and  from  1"  to  4"  long. 


FIG.  468. 


Fig.  470  is  a  stove-bolt,  from  £"  diameter  and  from  f"  to  3"  long. 

Figs.  471  and  472  are  machine-screws  without  nuts  ;  the  holes  in  the  metals 
are  tapped  to  receive  them  ;  Fig.  471  is  button-headed  ;  Fig.  472  a  counter- 
sunk head — both  slotted  to  admit  of  driving  by  a  screw-driver.  They  are 


M 


Fia.  470. 


FIG.  471. 


FIG.  472. 


T 
FIG.  473. 


FIG.  474. 


FIG.  475. 


made  of  various  sized  wire  and  lengths,  and  sold  by  the  gross  like  the  common 
wood-screw  (Fig.  473).  The  wood-screw  is  for  connecting  pieces  of  wood  to- 
gether, or  metal  to  wood.  They  are  of  very  great  variety,  usually  with  a 
gimlet-point,  so  that  they  can  be  driven  into  the  wood,  without  any  holes  being 
previously  made.  When  made  of  rods,  with  a  square  or  hexagonal  head  (Figs. 


16 


242 


MACHINE  DESIGN   AND  MECHANICAL  CONSTRUCTIONS. 


474  and  475)  to  admit  of  the  use  of  a  wrench,  they  are  called  lag-screws.  It 
will  be  seen  that  wood-screws  differ  in  their  thread  from  bolts  and  machine- 
screws.  The  thread  is  a  very  sharp  V,  flatter  on  the  upper  surface,  and  the 
flat  space  between  the  threads  wide  as  the  thread,  making  it  of  easier  introduc- 
tion into  the  wood,  and  retaining  as  much  strength  in  the  iron  as  in  the  wood. 
Fig.  476  is  a  stud-bolt,  which  is  screwed  firmly  into  one  of  the  pieces  of 
connected  metal ;  the  other  is  bored  so  as  to  slip  over  the  bolt,  and  the  nut 
then  brought  down  upon  it.  It  is  in  common  use  for  holding  on  the  bonnets 
of  steam-chests  and  water-chambers,  the  bolt  remaining  permanent. 


FIG.  476. 


FIG.  478. 


Fig.  477  is  a  hook-bolt ;  it  relieves  the  necessity  of  a  bolt  through  the  bot- 
tom-piece, and  may  be  turned  like  a  button,  to  loose  or  hold  the  bottom-plate. 

Fig.  478  is  another  kind  of  button -bolt ;  the  lower  end  can  revolve  on  a 
stud  or  pin  if  the  nut  be  raised  enough  to  clear  the  cap  or  upper  plate.  By 
this  arrangement  there  is  no  necessity  of  taking  off  the  nut  entirely ;  the  bolt 
lies  in  a  slot  in  the  cap,  and  the  nut  bears  on  three  sides. 


FIG.  479. 


FIG.  480. 


FIG.  481. 


Figs.  479,  480,  and  481  show  expedients  to  prevent  the  bolt  from  turning 
when  the  nut  is  screwed  on  or  off. 


FIG.  483. 


MACHINE  DESIGN   AND   MECHANICAL  CONSTRUCTIONS. 


243 


Fig.  482  is  an  anchor-bolt,  flattened  and  jagged,  introduced  into  a  hole  in 
masonry,  and  then  leaded  or  sulphured  in  ;  but  the  more  common  way  is  to 
split  the  lower  end  of  the  bolt,  insert  a  wedge  into  the  cleft,  place  the  bolt  in 
the  hole,  and  drive  the  wedge  in  against  the  bottom  of  the  hole,  thus  keying 
the  bolt  in  the  hole. 

Fig.  483  is  a  bolt  with  a  fang-nnt  or  corner  turned  down  and  driven  into 
the  wood  to  prevent  turning  ;  the  screwing  to  be  done  at  the  head. 

It  is  often  convenient  to  use  bolts  with  two  nuts,  as  in  Fig.  484,  or  collar- 
bolts,  which  are  readily  made  to  order,  and  of  any  dimensions. 

Fig.  485  is  a  hanger-bolt ;  the  lag- 
screw  part  is  screwed  into  the  wooden 
beam,  the  hanger  then  put  over  the  bolt, 
and  the  nut  put  on. 

Figs.  486  and  487  represent  forms  of 

turn-buckles,  and  the  swivel  and  pipe,  sometimes  designated  as  swivels.  Turn- 
buckles  are  very  useful  in  straining  tierods,  where  neither  end  of  the  bolts  can 
be  got  at.  By  turning  the  buckle,  the  rod  can  readily  be  made  longer  or 
shorter.  In  the  pipe-swivel,  rigid  and  left  threads  are  cut  on  the  bolts,  so 
that  each  turn  of  the  pipe  shortens  or  lengthens  the  tie  by  double  the  pitch  of 
the  screw.  The  turn-buckle  is  also  made  in  the  same  way,  with  two  screws 
instead  of  a  head  at  one  end. 


FIG. 


FIG.  485. 


FIG.  487. 


Screws,  unless  otherwise  ordered,  are  made  right-handed  ;  that  is,  turning 
the  nut  to  screw  up,  the  hand  moves  from  left  to  right,  the  apparent  motion  of 
the  sun. 

On  the  Strength  of  Bolts. — The  strength  of  a  bolt  depends  on  its  smallest 
section — that  is,  between  the  bottom  of  the  threads.  It  is  very  common, 
therefore,  especially  in  long  bolts,  to  upset  the  screw-end,  so  that  the  screw  may 
be  cut  entirely  from  this  extra  boss,  or  re-enforce.  Bolt-ends  (Fig.  488)  are 
sold  either  with  or  without  re-enforce,  to  be  welded  to  bolts.  It  will  be 
observed  that  the  ends  of  the  pipe-swivel  bolts  (Fig.  487)  are  thus  upset. 


FIG.  488 

In  the  following  table,  the  sizes  and  dimensions  "of  bolts  and  nuts  are  from 
the  United  States  standard,  and  the  strength,  or  safe-load  of  the  bolts,  is 
computed  from  the  report  of  the  committee  on  the  test  of  wrought-iron  and 
chain-cables  to  the  United  States  Government  in  1879.  Nuts  and  heads  as 
furnished  are  either  hexagonal  or  square.  Columns  4,  5,  and  6  apply  equally 
to  either. 


244  MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


Diameter  of 
screw  in 
inches. 

Diameter  at 
root  of 
thread. 

Thread  per 
inch  of 
length. 

Short  diam- 
eter of  nut 
and  head. 

Thickness  of 
nut. 

Thickness  of 
head. 

Safe-load  of 
upset  bolts. 

Safe-load  of 
plain  bolts. 

i 

TV 

i 

A 

f 

•185 

20 

3 

TIT 

if 

t 

•240 
•294 

18                if 
16                ii 

t 

H 

1,700 

P 

•344 
•400 

13      r 

TV 
i 

i! 

TV 

1,900 

2,200 

TV 

•494 

12                f£ 

TV 

If 

2,500 

f 

•507 

11 

1A 

f 

2,800 

f* 

•620 

10 

H 

ii 

ii 
f 

If 
Is 

6,000 

3,200 
3,600 

I1 

•731 

9 

Hi 
1A 

H 

Ii 

7,000 
8,000 

4,300 
5,100 

1 

•837 

8 

if 

i 

ii 

10,000 

7,000 

li 

•940 

7 

lit 

il 

14 

12,000 

9,000 

li 

1-065 

7 

2 

il 

i 

15,000 

11,000 

H 

1-160 

6 

2A 

if 

iA 

18,000 

13,500 

li 

1-284 

6 

2f 

IA 

21,000 

16,000 

If 
If 

1-389 
1-490 

51 

5 

It 

if 
if 

|r 

24,000 
28,000 

19,000 
22,300 

11 

1-615 

5 

ii 

32,000 

25,500 

2 

1-712 

4| 

8f 

2 

IA 

36,000 

29,300 

21 

8  A 

21 

114 

40,000          33,000 

2i 

1-962 

4* 

81 

If 

45,000 

37,000 

2f 

3ii 

2f 

iff 

50,000 

41,500 

2i 

2-175 

4 

31 

21 

55,000 

46.000 

2f 

4rV 

2f 

2^L 

2f 

2-425 

4 

41 

2f 

2i 

21 

4A 

21 

2^2 

3 

2-629 

81 

4| 

3 

^A 

81 

2-879 

81 

5 

31 

2i 

81 

3-100 

31 

5f 

3i 

2^  \ 

3 

3-317 

3              of 

3f 

21 

4 

3-567 

3 

61 

4 

8A 

41 

3-798 

21 

6i 

41 

81 

41 

4-028 

2f 

61 

41 

8A 

4f 

4-255 

2f 

71 

4f 

8f 

5 

4-480 

2i 

*jf 

5 

51 

4-730 

2i 

8 

51 

4lb 

51 

5-058              2| 

8f 

51 

4T\ 

5f 

5-203 

2f 

8f 

5f 

H 

6 

5-423              21 

6 

Washers  (Fig.  489) — in  common  use  to  provide  seatings  for  nuts  which 
would  otherwise  rest  on  rough  metallic  surfaces,  and  also  to  adapt  bolts  to 
shorter  spaces  than  their  lengths— are  sold  for  bolts  up  to  2"  diameter.  Cir- 


FIG.  489. 


FIG.  490. 


MACHINE   DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


245 


cular  in  form,  their  diameter  is  slightly  in  excess  of  that  of  the  largest  diam- 
eter of  the  nut,  and  the  hole  that  of  the  bolt,  and  thickness  from  •£$"  to  -J-", 
according  to  the  diameter  of  the  bolt.  The  square  washer  is  used  under  both 
head  and  nut  on  surfaces  of  wood,  and  of  dimensions  suited  to  the  stress. 
That  they  may  neither  sink  into  the  wood,  nor  bend  or  break,  cast-iron  is  fre- 
quently used,  and  often,  as  shown  in  Fig.  490,  for  roof-frames. 

Shafts  and  Axles. — Short  shafts,  revolving  in  bearings  or  boxes,  or  fastened 
with  pulleys,  drum,  or  wheels  revolving  on  them,  are  called  axles  ;  but  shafts 
of  large  dimension  or  extent,  and  revolving,  are  usually  termed  shafts,  as  water- 
wheel  shafts  and  fly-wheel  shafts.  These  may  be  independent  ;  that  is,  a  sin- 
gle shaft,  revolving  in  its  bearings,  or  they  may  be  coupled  together,  forming 
what  is  termed  a  line  of  shafting.  The  small  shafts,  as  in  clock-work  and 
spinning-machinery,  are  termed  pins  and  spindles. 

Shafts  and  axles  are  made 
of  wood  and  metal,  and  of  va- 
ried sections  and  form. 

Wooden  shafts  are  polygo- 
nal, circular,  or  square  section  FIG.  491. 
(Fig.  491). 

Wrought  metal,  iron,  or  steel  shafts,  are  almost  invariably  circular  in  sec- 
tion, but  sometimes  square. 

Cast-iron  is  used  in  great  variety  of  section  and  form  for  shafts  (Fig.  492)  ; 
without   uniformity  longi- 
tudinally, but   adapted  to 
their  position  and  load. 

Formerly,   either   wood 
or  cast-iron  was  invariably 

used  for  water-wheel  shafts  ;  FIG.  492. 

but    a   change   of   motors, 

from  the  breast,  over-shot  and  under-shot  wheels  to  reactors  or  turbines,  has 
involved  an  entire  change  of  construction,  and  now  only  wrought-iron  is  used. 
Still,  wooden  shafts  are  often  used  in  machines  subject  to  wet  or  shock,  and 
often  from  greater  convenience  in  procuring  the  material  ;  and,  from  the  same 
cause,  the  bearings  or  bushings  on  which  the  shafts  revolve  are  of  the  same 
material,  and  serve  a  good  purpose  where  the  movements  are  not  continuous 
or  rapid.  But  it  is  usual  to  make  metal  boxes,  in  which  the  rounded  ends  of 
shafts  revolve  ;  these  ends  are  called  journals  or  gudgeons.  The  diameters 
and  lengths  of  journals  depend  on  the  weight  to  be  supported,  the  material  of 
shafts  and  bearings,  and  the  velocity  at  which  the  shafts  are  run. 

TABLE  OF  DIAMETER  OF  JOURNALS  FOE  HEAVY  WORK. 


Total  load  in 

DIAMETER    IN    INCHES. 

Total  load  in 

DIAMETEK   IN   INCHES. 

Total  load  in 

DIAMETER  IN  INCHES, 

pounds. 

Cast- 

Wrought- 

pounds. 

Cast- 

Wrought- 

pounds. 

Cast- 

Wrought- 

iron. 

iron. 

iron. 

iron. 

iron. 

iron. 

1,100 

2 

1-7 

30,000 

6 

5-1 

137,000 

10 

8'6 

3,700 

3 

2-5 

44,000 

7 

6-0 

183,000 

11 

9-4 

8,800 

4 

3-4 

70,000 

8 

6-9 

237,000 

12 

10'3 

17,000 

5 

4-3 

100,000 

9 

7-7 

312,000 

13 

11-2 

246 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


PS. 


FIG.  493. 


FIG.  494. 


The  usual  length  of  journals  is  from  once  to 
twice  their  diameters,  but  this  is  to  be  modified 
by  the  speed  at  which  the  shafts  are  run  ;  if 
slow-moving,  one  diameter  in  length  is  ample, 
but  at  very  high  speed,  and  small  shafts,  like 
those  of  circular  saws,  from  4  to  6  diameters  is 
not  uncommon.  If  the  boxes  are  of  cast-iron, 
they  will  sustain  the  load  given  in  the  above 
table  for  large  journals ;  but  when  the  boxes 
are  lined  with  brass  and  composition,  or  Babbit- 
metal,  the  first  should  not  be  loaded  beyond  500 
pounds  per  square  inch,  on  half-circumferential 
section,  or  750  pounds  on  the  axial  section.  Bab- 
bit-metal should  have  a  somewhat  less  load,  say 
500  pounds  on  the  axial  section. 

Wooden  shafts  are  sometimes  fitted  with  wood- 
en journals  and  boxes,  but  the  usual  practice  is 
to  insert  cast-iron  journals. 

Figs.  493  and  494  represent  different  views  of 
a  wooden  shaft.  Fig.  493  shows  at  one  end  the 
side  elevation  of  the  shaft,  furnished  with  its  iron 
ferules  or  collars  and  its  gudgeon  ;  at  the  other 
end,  the  shaft  is  shown  in  sections,  giving  the 
ferules  in  section,  but  showing  the  central  spin- 
dle with  its  feathers  in  an  external  elevation. 
Generally,  in  longitudinal  sections  of  objects  in- 
closing one  or  more  pieces,  the  innermost  or  cen- 
tral piece  is  not  sectioned  unless  it  has  some  in- 
ternal peculiarity,  the  object  of  a  section  being 
to  show  and  explain  peculiarities,  and  being  there- 
fore unnecessary  when  the  object  is  solid  ;  on 
this  account,  bolts,  nuts,  and  solid  cylindrical 
shafts  are  seldom  drawn  in  section.  Fig.  494  is 
an  end  view  of  the  shaft,  showing  the  fitting  of 
the  spindle  B  and  its  feathers  into  the  end  of 
the  shaft,  and  the  binding  of  the  whole  by  ferules 
or  hoops,  a  a.  The  spindles  B,  which  are  let 
into  the  ends,  are  cast  with  four  feathers  or 
wings,  c.  The  tail-piece  ~b  is  by  most  millwrights 
omitted.  The  ends  of  the  beam  are  bored  for 
the  spindle,  and  grooved  to  receive  the  feathers  ; 
the  casting  is  then  driven  into  its  place,  hooped 
with  hot  ferules,  and  after  this  hard-wood  wedges 
are  driven  in  on  each  side  of  the  feathers,  and 
iron  spikes  are  sometimes  driven  into  the  end  of 
the  wood. 

Figs.   495,  496,   and  497  represent  different. 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


247 


FIG.  496. 


views  of  a  cast-iron  shaft  of  a  water-wheel.  Fig. 
495  is  an  elevation  of  the  shaft,  with  one  half  in 
section  to  show  the  form  of  the  core  ;  Fig.  496,  an 
end  elevation  ;  Fig.  497,  a  section  on  the  line  c  c 
across  the  center.  The 
body  is  cylindrical  and  hol- 
low, and  is  cast  with  four 
feathers,  c  c,  disposed  at 
right  angles  to  each  other, 
and  of  an  external  para- 
bolic outline.  Near  the 
extremities  of  these  feath- 
ers four  projections  are 
cast,  for  the  attachment  of 
the  bosses  of  the  water- 
wheel.  These  projections 

are  made  with  facets,  so  as  to  form  the  corners  of  a 
circumscribing  square,  as  shown  in  Fig.  496,  and 
they  are  planed  to  receive 
the  keys  by  which  they  are 
fixed  to  the   naves  which 
are  grooved  to  receive  them. 
The  shaft  is  cast  in  one  en- 
tire piece,  and  the  journals 
are  turned. 

It  will  be  observed  that 
although  no  weight  is  sup- 
ported at  the  center,  yet 
there  is  an  increase  in  the 
diameter  of  the  feathers  at 

this  line  ;  the  weight  of  the  shaft  itself  is  a  consid- 
erable factor. 

Fig.  498  represents  the  section  of  a  portion  of  a 
breast  water-wheel,  with  a  cast-iron  shaft,  formerly 
much  in  use  in  this  country,  in  which  stiffness  was 
given  to  the  wheel  and  shaft  by  wooden  trusses. 
These  shafts  are  cast  circular,  in  two  lengths,  con- 
nected at  the  center,  with  circular  bosses  on  which 
the  naves  of  the  wheel  are  keyed. 

Journals  of  independent  shafts  are  always  of  less 
diameters  than  those  of  the  rest  of  the  shafts,  and  if 
the  load  on  each  is  nearly  equal  the  diameters  of  the 
two  journals  are  equal ;  but,  if  the  load  is  not  cen- 
tral, the  diameters  are  proportioned  to  their  several 
loads,  as  shown  on  pages  228,  229. 

Shafts  of  wrought-iron,  of  less  diameter  than  six 
inches,  are  fitted  by  turning  down  the  journals 


FIG.  497. 


FIG.  495. 


248 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


only.  Large  shafts  are  generally 
forged,  mostly  in  steps,  as  in  Fig. 
499,  with  the  largest  boss  beneath 
the  gear  or  pulley-hub,  and  suffi- 
ciently above  the  next  boss,  on  each 
side,  to  admit  of  the  planing  or  cut- 
ting of  the  key-seats. 

To  determine  the  size  of  a  shaft, 
considered  as  a  beam  merely,  but 
with  a  shafting  load — as  by  the  rev- 
olution of  the  shaft — each  longitu- 
dinal line  of  surface  has  to  undergo 
successively  tension  and  compres- 

FlG  498  sion.      The  safe  load  of  wrought- 

iron  is  estimated  at  6,000  pounds 

per  square  inch,  and  the  formula  on  which  the  graphic  diagram  (Fig.  500)  is 
constructed  is  d  =  '06  |/  w  I,  d  being  diameter,  I  =  length  between  bearings, 


FIG.  499. 


l)oth  in  inches,  w  the  load  in  pounds  ;  the  load  is  not  only  the  weight  of  shaft 
and  pulleys  or  gears,  but  also  the  stress  in  transmitting  the  power. 

Use  of  Diagram. — Suppose  w  =  50,000  pounds,  and  I  =  6  feet  =  72",  then 


14' 


13 


12 


111 

I 


10 


56789 
Product  of  Load  and  Span  in  Millions. 
FIG.  500. 


10 


12,000,OuO 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


249 


w  I  =  3,600,000,  the  ordinate  of  3 '6  cuts  the  curve  on  the  abscissa  9  '2,  which  is 
the  required  diameter  of  the  shaft  in  inches. 


FIG.  501. 


FIG.  502. 


FIG.  503. 


FIG.  504. 


Keys  are  pieces  of  metal,  usually  steel,  employed  to  secure  the  hubs  of  pul- 
leys, gears,  and  couplings  to  shafts.     They  may  be  sunk  keys  (Fig.  501),  flat 
keys  (Fig.  502),  and  hollow  keys  (Fig.  503).     The  shaded 
circle  represents  the  shaft.     The  breadth  of  the  key  (Fig. 
504)  is  uniform,  but  the  thickness  is  tapered  about  one 
eighth  of  an  inch  per  foot.     The  shoulder  h  is  for  the  pur- 
pose of  drawing  out  the  key.     Sunk  keys  are  not  necessa- 
rily taper.     Some  prefer  them  of  uniform  section,  and  to 
force  the  hub  on  over  the  key. 

PROPOKTIONS  OF  SUNK  KEYS. 


DIAMETER  OF  SHAFT,    IN  INCHES. 

1 

2 

3 

4 

5 

6 

Breadth  of  key.  .      

o 

4 

1 

1* 

j5 

Thickness  of  key 

•25 

'34 

'43 

•52 

•61 

'71 

Depth  sunk  in  shaft    

•10 

•125 

'15 

•175 

•20 

•225 

Depth  sunk  in  wheel  

•15 

•215 

•28 

'345 

•41 

•485 

DIAMETEE  OF  SHAFT,   IN  INCHES.         7 

8 

9 

10 

11 

12 

Breadth  of  key                                               1£ 

01 

O8 

O5 

9i 

qi 

Thickness  of  key    .         ....                   i  '80 

^$ 
•89 

*f 

•98 

*f 

*¥ 

1'lfi 

«*t 
1  -9fi 

Depth  sunk  in  shaft  -25 

•275 

•30 

'325 

'35 

•  Q'JK 

Depth  sunk  in  wheel  -55 

•615 

•68 

•745 

'81 

'875 

Car- Axles. — Fig.  505  is  the  form  and  dimensions  of  axle 
adopted  as  standard  by  the  American  Master  Car-Builders' 
Association  for  wrought-iron  and  steel. 

Shafting. — Thus  far,  independent  shafts  or  axles  have 
been  treated  of,  and  the  dimensions  have  been  established 
mostly  by  the  load  acting  transversely;  but,  in  transfer- 
ring power  to  machines,  lines  of  shafting  are  necessary, 
almost  invariably  of  wrought-iron  or  steel  bars,  which  are 
subject  not  only  to  transverse  but  also  torsional  stress. 
When  there  are  no  pulleys  or  gears  on  the  shafts  between 
the  bearings,  and  the  couplings  are  close  to  the  bearings, 
there  is  still  an  amount  of  deflection  due  to  the  weight  of 
the  shaft.  James  B.  Francis,  C.  E.,  puts  the  maximum 
distances  between  bearings  for  shafts  of  wrought-iron  or 
steel,  under  these  conditions,  as  follows  : 


SE 

1  

7 

f 

V 

r2 

^ 

^-> 

.*. 

•: 

•r- 

i 

i 

^ 

.-ji  * 

«3 

250 


MACHINE   DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


Diameter        Distance  between    Diameter    Distance  between    Diameter        Distance  between 


of  shaft. 
1" 

2 
3 
4 


bearings. 

12ft. 
15 
18 
20 


of  shaft. 
5* 
6 

7 
8 


bearings. 

21  ft. 
22 
24 
25 


of  shaft. 

9" 
10 
11 

12 


bearings. 

26ft. 

27 
28 
28 


The  diagram  (Fig.  506)  is  one  established  by  J.  T.  Henthorn,  M.  E.  of  the 
Corliss  Steam-Engine  Company,  to  determine  the  size  of  wrought-iron  shaft- 
ing, to  transmit  a  fixed  amount  of  horse-power. 

400 


Horse-Power. 
FIG.  506. 

Use  of  Table. — To  find  the  size  of  a  shaft  making  150  revolutions,  and  trans- 
mitting 350  horse-power. 

The  intersection  of  the  ordinate  of  350  with  the  abscissa  of  150  is  between 
the  diagonals  5  and  5£,  and  the  diameter  of  the  shaft  may  be  taken  safely  at  5J". 

Mr.  Francis  has  constructed  a  table  from  his  own  experiments,  of  which 
the  following  is  a  synopsis  : 

''  The  following  table  gives  the  power  which  can  be  safely  carried  by 
shafts  making  100  revolutions  per  minute.  The  power  which  can  be  carried 
by  the  same  shafts  at  any  other  velocity  may  be  found  by  the  following  simple 
rule  : 

"Multiply  the  power  given  in  the  table  by  the  number  of  revolutions  made 
by  the  shaft  per  minute  ;  divide  the  product  by  100  ;  the  quotient  will  be  the 
power  which  can  be  safely  carried." 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


251 


Horse-power  which  can  be  safely  transmitted  by  shafts  making  100  revolutions  per  minute,  in  which  the  trans- 
verse strain,  if  any,  need  not  be  considered ;  if  of 


Diameter 
in  inches. 

Wrought- 
iron. 

Steel. 

Diameter 
in  inches. 

Wrought- 
iron. 

Steel. 

Diameter 
in  inches. 

Wrought- 
iron. 

Steel. 

1- 

2-0 

32 

4'5 

182- 

291- 

7-5 

843' 

1350- 

1-5 

6-7 

10-7 

5- 

250- 

400' 

8' 

1024' 

less- 

2- 

16-0 

25-6 

5'5 

332- 

532' 

8'5 

1228- 

ees- 

2'5 

31-2 

50" 

6' 

432' 

691" 

9- 

1458' 

2332' 

3- 

54-0 

86-4 

6'5 

549' 

878- 

9-5 

1714- 

2743' 

3-5 

85'7 

137- 

7- 

686' 

1097' 

10 

2000" 

3200' 

4' 

128- 

204' 

The  diagram  and  table  given  are  applicable  to  shafts  which  are  called  sec- 
end  movers,  subject  to  no  sudden  shock.  For  first  movers,  Mr.  Francis  takes 
but  one  half  the  horse-power  given  in  the  table  for  any  diameter  of  shafts. 
Of  late,  cold-rolled  shafts  can  be  procured  in  the  market,  which  are  much 
stiffer  than  turned  shafts,  but  not  equal  to  that  given  for  steel  in  the  table. 

It  is  usual  to  make  the  shafts  of  second  and  third  movers  throughout  manu- 
factories and  shops  of  uniform  diameter,  without  reduction  at  the  journals,  the 
end-slip  being  prevented  by  collars  keyed  or  fastened  by  set-screws.  The  usual 
length  between  bearings  is  from  7  to  10  feet ;  but  that  they  may  run  smooth, 
and  not  spring  intermediately,  it  is  desirable  that  they  should  never  be  less 
than  2  inches  diameter,  and  that  the  pulleys  or  gears  through  which  the  power 
is  transmitted  to  the  next  mover  or  to  the  machine  should  be  as  near  as  pos- 
sible to  the  bearing. 

Fig.  507  represents  a  line  of  shafting.  A  is  an  upright  shaft ;  a  a,  bevel- 
gears  ;  b  b,  bearings  for  the  shafts  ;  c,  coupling  or  connection  of  the  several 
pieces  of  shafting.  These  shafts  are  intended  to  be  of  wrought-iron.  No  re- 
duction is  made  for  the  journal,  no  bosses  for  pulleys  or  gears.  As  the  power 
is  distributed  from  this  line  of  shafting,  the  torsional  strain  diminishes  with 


FIG.  507. 

the  distance  from  the  bevel-gears  or  first  movers,  and  the  diameter  of  each 
piece  of  shafting  may  be  reduced  consecutively,  if  necessary ;  but  uniformity 
will  generally  be  found  to  be  of  more  importance  than  a  small  saving  of  iron. 
The  drawing  given  is  of  a  scale  large  enough  to  order  shafting  by,  but  the 
dimensions  should  be  written  in. 

In  laying  out  lines  of  shafting,  the  position  of  the  bearings  is  usually  fixed, 
and  the  lengths  of  shafts  must  be  determined  thereby,  with  as  few  couplings 
as  possible.  When  there  is  no  necking  or  reduction  of  the  shafts,  which  is 
usually  the  case,  the  orders  given  for  shafting  will  be  so  many  lengths  and  of 
such  diameters,  and  so  many  couplings  and  hangers.  When 'there  is  to  be  a 


252 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


necking,  the  sketch  for  the  order  may  be  very  simple,  showing  length  and 
diameter  of  shaft,  and  position,  length,  and  diameter  of  bearing. 

The  joints  or  couplings  are  generally  made  near  the  bearings,  and  it  is  also 
usual  to  bring  the  pulleys  as  near  the  bearings  as  possible.  It  frequently  hap- 
pens, therefore,  that  the  coupling  and  pulley  are  needed  at  the  same  point ; 
to  remedy  this,  as  the  position  of  the  pulley  depends  on  the  machine  which  it 
is  required  to  drive,  it  frequently  can  not  be  moved  without  considerable  in- 
convenience or  loss  of  room ;  the  shaft  will  have,  therefore,  to  be  lengthened 
or  shortened,  to  change  position  of  coupling  ;  or,  if  the  couplings  are  plate 
couplings,  the  coupling  and  pulley  may  be  made  together. 

When  a  horizontal  shaft  is  supported  from  beneath,  its  bearing  is  usually 
called  &  pillow-  or  plumber --block,  or  standard ;  if  suspended,  the  supports  are 
called  hangers. 


FIG.  509. 

Figs.  508  and  509  are  the  elevation  and  plan  of  a  pillow-block.  It  consists 
of  a  base  plate,  A,  the  body  of  the  block  B,  and  the  box  C.  The  plate,  as  in 
the  step,  is  bolted  securely  to  its  base,  the  surface  on  which  the  block  B  rests 
being  horizontal.  A  and  B  are  connected  by  bolts  passing  through  oblong 
holes,  so  as  to  adjust  the  position  in  either  direction  laterally.  The  box  or 
bush  C  is  of  brass,  in  two  parts  or  halves,  extending  through  the  block,  and 
forming  a  collar  by  which  it  is  retained  in  its  place.  The  cap  of  the  block  is 
retained  by  the  screws  o  o  o  ;  in  the  figure  there  are  two  screws  on  one  side 
and  one  on  the  other ;  often  four  are  used,  two  on  each  side,  but  most  fre- 
quently but  one  on  each  side. 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS.  253 

F 


FIG.  511. 


The  standard  is  simply  a  modification  of  the  pillow-block,  being  employed 
for  the  support  of  horizontal  shafts  at  a  considerable  distance  above  the  founda- 
tion-plate. Fig.  510  is  a  front  elevation ;  Fig.  511,  a  plan  ;  and  Fig.  512,  an 
end  elevation  of  a  standard.  Like  the  pillow-block,  the  plate  A  is  fastened  to 


254: 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


the  foundation  itself,  and  the  upper  surface  is  placed  perfectly  level  in  both 
directions.  On  these  bearing  surfaces,  a  a  a,  the  body  of  the  standard  rests,  and 
can  be  adjusted  in  position  horizontally,  and  then  clamped  by  screws  to  the 
foundation-plate,  or  keyed  at  the  ends. 

Elevations  and  plan  are  usually  drawn  in  such  positions  to  each  other  that 
lines  of  construction  can  be  continued  from  one  to  the  other,  which  not  only 
simplifies  the  drawings,  but  makes  them  more  readily  intelligible.  Letters  and 
dotted  lines  in  these  figures  illustrate  this  sufficiently. 

It  will  be  observed  that  the  sides  of  the  elevations  are  represented  as  broken; 
this  is  often  done  in  drawing,  when  the  sides  are  uniform,  and  ecpnomy  of 
space  on  the  paper  is  required. 

Suspended  bearings  or  hangers  for  horizontal  shafts  are  divided  into  two 
general  classes,  side-hangers  (Figs.  513,  514)  and  sprawl-hangers ;  the  figures 
will  sufficiently  explain  the  distinction.  The  side-hanger  is  the  more  conven- 
ient when  it  is  required  to  remove  the  shaft,  and  when  the  strain  is  in  one 


FIG.  513. 


FIG.  514. 


direction,  against  the  upright  part ;  they  are  generally  used  for  the  smaller 
shafts,  but  sprawl-hangers,  affording  a  more  firm  support  in  both  directions, 
are  used  as  supports  for  all  the  heavier  shafts.  Hangers  are  bolted  to  the  floor- 
timbers,  or  to  strips  placed  to  sustain  them,  the  centers  of  the  boxes  being 
placed  accurately  in  line,  both  horizontally  and  laterally. 

Fig.  515  represents  the  elevation  of  a  sprawl-hanger ;  Fig.  516,  the  plan 
looking  from  above,  with  cover  of  box  off  ;  Fig.  517,  a  section  on  the  line  A  B, 
Fig.  515. 

Fig.  518  represents  the  elevation  of  a  bracket,  or  the  support  of  a  shaft 
bolted  to  an  upright ;  the  box  is  movable,  and  is  adjusted  laterally  by  the  set- 
screws.  Fig.  519  is  a  front  elevation  of  the  back  plate  cast  on  the  post ;  it  will 
be  seen  that  the  holes  are  oblong,  to  admit  of  the  vertical  adjustment  of  the 
bracket. 

Figs.  520-523  represent  different  views  of  what  may  be  called  a  yoke-hanger. 


MACHINE  DESIGN  AND   MECHANICAL  CONSTRUCTIONS.  255 


FIG.  517 


/ 

i 

i 
1 

/ 

5 

0 

A 

r 

o 

i 

3 

c 

J 

1 

7 

~v 

FIG.  510. 


Fig.  520  is  a  front  and  Fig.  521  a  side  elevation;  Fig.  522  a  plan  of  the  hanger, 
looking  up  ;  and  Fig.  523  a  plan  of  the  yoke,  looking  down  upon  it.  A  is 
the  plate  which  is  fastened  to  the  beam,  E  is  the  yoke,  and  B  the  stem  of  the 


0 

0 

0 

0 

256  MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


FIG.  520. 


FIG.  521. 


FIG.  522. 


FIG.  523. 


yoke,  cut  with  a  thread  so  as  to  admit  of  a  vertical  adjustment ;  the  box  D  of 
the  shaft  C  is  supported  by  two  pointed  set-screws  passing  through  the  jaws  of 
the  yoke  ;  this  affords  a  very  flexible  bearing,  and  a  chance  for  lateral  adjust- 
ment. 

For  Upright  Shafts.— Footstep,  or  Step,  for  an  Upright  Shaft.—  Fig.  524 
represents  an  elevation,  Fig.  525  a  plan  of  the  step.  It  consists  of  a  founda- 
tion or  bed-plate,  A.  a  box,  B,  and  a  cap  or  socket,  C.  The  plate  A  is  firmly 
fastened  to  the  base  on  which  it  rests  ;  in  the  case  of  heavy  shafts,  often  to  a 
base  of  granite.  The  box  B  is  placed  on  A,  the  bearing  surface  being  accu- 
rately leveled,  and  fitted  either  by  planing  or  chipping  and  filing  ;  h,  b,  #,  are 
what  are  commonly  called  chipping-pieces,  which  are  the  bearing  surfaces  of 
the  bottom  of  B.  A  and  B  are  held  together  by  two  screws  ;  the  holes  for 
these  are  cut  oblong  in  the  one  plate  at  right  angles  to  those  of  the  other ;  this 
admits  of  the  movement  of  the  box  in  two  directions  to  adjust  nicely  the  lat- 
eral position  of  the  shaft,  after  which,  by  means  of  the  screws,  the  two  plates 
are  clamped  firmly  to  each  other.  0,  the  cup  or  bushing,  which  should  be 
made  of  brass,  slips  into  a  socket  in  B.  Frequently  circular  plates  of  steel  are 
dropped  into  the  bottom  of  this  cup  for  the  step  of  the  shaft.  The  cup  0,  in 


MACHINE  DESIGN   AND  MECHANICAX  CONSTKUCTIONS.  257 


FIG.  525. 


case  of  its  sticking  to  the  shaft,  will  revolve  with  the  shaft  in  the  box  B ;  if 
plates  are  used,  these  also  admit  of  movement  in  the  cup. 

Fig.  526  represents  the  elevation  of  a  bearing  for  an  upright  shaft,  in  which 


FIG.  526. 

the  shaft  is  held  laterally  by  a  box  and  bracket  above  the  step.  The  step  B  is 
made  larger  than  the  shaft,  so  as  to  reduce  the  amount  of  wear  incident  to  a 
heavy  shaft.  The  end  of  the  shaft  and  the  cup  containing  oil  are  shown  in 


258 


MACHINE   DESIGN   AND   MECHANICAL  CONSTRUCTIONS. 


dotted  line.     The  bed-plate  A  rests  on  pillars,  between  which  is  placed  a  pil- 
low-block or  bearing  for  horizontal  shaft. 

Figs.  527  and  528  represent  the  elevation  and  vertical  section  of  the  suspen- 
sion bearing  used  by  Mr.  Boy  den  for  the  support  of  the  shaft  of  his  turbine- 
wheels.  It  having  been  found  difficult  to  supply  oil  to  the  step  of  such  wheels, 
it  was  thought  preferable  by  him  to  suspend  the  entire  weight  of  wheel  and 
shaft,  where  it  could  be  easily  attended  to.  The  shaft  (see  section)  is  cut  into 


FIG.  527. 


FIG.  528. 


necks,  which  rest  on  corresponding  projections  cast  in  the  box  ~b  ;  the  spaces  in 
the  box  are  made  somewhat  larger  than  the  necks  of  the  shaft,  to  admit  of  Bab- 
bitting, as  it  is  termed,  the  box  ;  that  is,  the  shaft  being  placed  in  its  position 
in  the  box,  Babbitt,  or  some  other  soft  metal  melted,  is  poured  in  round  the 
shaft,  and  in  this  way  accurate  bearing  surfaces  are  obtained  ;  projections  or 
holes  are  made  in  the  box  to  hold  the  metal  in  its  position.  The  box  is  sus- 
pended by  lugs  b,  on  gimbals  c,  similar  to  those  used  for  mariners'  compasses, 


i 


FIG.  529. 


FIG.  530. 


which  give  a  flexible  bearing,  so  that  the  necks  may  not  be  strained  by  a  slight 
sway  of  the  shaft.  The  screws  e  e  support  the  gimbals,  consequently  the  shaft 
and  wheel ;  by  these  screws  the  wheel  can  be  raised  or  lowered,  so  as  to  adjust 
its  position  accurately ;  beneath  the  box  will  be  seen  a  movable  collar,  to  adjust 
the  lateral  position  of  shafts. 


MACHINE   DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


259 


Figs.  529  and  530  are  the  plan  and  elevation  for  the  step,  or  rather  guide 
{as  it  bears  no  weight),  of  the  foot  of  the  shaft  of  these  same  turbines.  The 
plate  A  is  firmly  bolted  to  the  floor  of  the  wheel-pit ;  the  cushions  C,  holding 
the  shaft,  are  either  wooden  or  cast-iron,  and  admit  of  lateral  adjustment  by 
the  three  rows  of  set-screws. 
In  construction,  the  hanger  and 
guide  of  Mr.  Boyden  were  found 


FIG.  531. 


to  be  too  expensive,  and  wooden 
steps  (Fig.  531)  are  now  almost  FIG.  532. 

universally   used   for   turbines. 

They  are  made  either  conical  or  a  portion  of  a  sphere,  of  various  woods,  usually 
lignum-vitae,  but  oak  and  poplar  are  preferred  by  some.  The  load  is  from 
fifty  to  seventy-five  pounds  per  square  inch.  The  fibers  of  the  wood  are 
placed  vertically,  and  afford  a  very  excellent  bearing  surface.  Water  is  some- 
times introduced  into  the  center  of  the  wood,  or  into  a  box  around  it,  from  the 
upper  level  of  water.  When  cast-iron  or  steel  is  used  for 
the  step,  it  is  usual  to  incase  the  box  and  supply  oil  by  lead- 
ing a  pipe,  sufficiently  high  above  the  surface  of  the  water, 
to  force  the  oil  down. 


FIG.  534. 


For  long,  upright  shafts,  it  is  very  usual  to  suspend  the  upper  portion  by  a 
suspension-box,  and  to  run  the  lower  on  a  step,  connecting  the  two  portions 
by  a  loose  sleeve  or  expansion  coupling,  to  prevent  the  unequal  meshing  of  the 


260 


MACHINE   DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


bevel- wheels,  incident  to  an  alteration  of  the  length  of  shaft  by  variations  of 
temperature.  The  suspension  is  frequently  made  by  a  single  collar  at  the  top 
of  the  shaft. 

Figs.  532  and  533  are  perspectives  of  the  hangers  made  by  William  Sellers 
&  Co.,  of  Philadelphia  ;  and  Fig.  534  a  section  showing  the  adjustment  of  the 
boxes  in  Fig.  532. 

The  boxes  are  of  cast-iron,  long  in  proportion  to  the  diameters  of  shafts  ; 
the  center  bearings  are  spherical,  and  are  adjusted  in  position  vertically  by  the 
screws  d  and  e  ;  b  I  are  cups,  to  contain  grease,  which  will  melt  if  the  bearings 
become  heated,  but  the  lubrication  depends  on  an  oil-cup  dripping  oil  into  the 
center  of  the  bearing ;  /  is  a  cast-iron  drip-pan  to  catch  the  waste  oil  from  the 
journal. 

Fig.  533  is  a  view  of  side  hanger  adapted  to  a  counter  shaft,  and  the  square 
slot  a  is  for  the  shipping-bar.  This  form  of  hanger  is  more  common  than 
that  shown  in  Figs.  513  and  514  ;  the  cap  in  this  last  is  held  down  by  a  wedge, 
in  Fig.  533,  by  a  lateral  screw  ;  but  with  most  makers  the  screw  is  vertical, 
clamping  the  cap  to  the  lower  part  of  the  box. 

Couplings  are  the  connections  of  shafts,  and  are  varied  in  their  construction 
and  proportions  often  according  to  the  mere  whim  of  the  mechanic  making 
them. 

The  Face  Coupling  (Fig.  535)  is  the  one  in  most  general  use  for  the  con- 
necting of  wrought-iron  shafts  ;  it  consists  of  two  plates  or  disks  with  long, 

strong  hubs,  through  the  center  of 
which  holes  are  accurately  drilled 
to  fit  the  shaft  ;  one  half  is  now 
drawn  on  to  the  shaft,  and  tightly 
keyed ;  the  plates  are  faced  square 
with  the  shaft,  and  the  two  faces 
are  brought  together  by  bolts.  The 
number  and  size  of  the  bolts  depend 
upon  the  size  of  the  shaft ;  never 
FIG.  535.  less  than  4  for  shafts  less  than  3 

inches  diameter,  and  more  as  the 

diameter  increases  ;  the  size  of  the  bolts  varies  from  f  to  1£  inch  in  diameter. 
The  figure  shows  a  usual  proportion  of  parts  for  shafts  of  from  2  to  5  inches 
diameter  ;  for  larger  than  these,  the  proportion  of  the  diameter  of  the  disk  to 
that  of  the  shaft  is  too  large. 

Fig.  536  is  a  rigid  sleeve  coupling  for  a  cast-iron  shaft ;  it  consists  of  a  solid 
hub  or  ring  of  cast-iron  hooped  with  wrought-iron  ;  the  shafts  are  made  with 
bosses,  the  coupling  is  slipped  on  to  one  of  the  shafts,  the  ends  of  the  two  are 
then  brought  together  ;  the  coupling  is  now  slipped  back  over  the  joint,  and 
firmly  keyed.  This  is  an  extremely  rigid  connection.  Some  makers  use  keys 
without  taper,  and  force  the  couplings  on  the  shafts. 

Fig.  537  is  a  screw  coupling  for  the  connecting  of  the  lighter  kinds  of  shafts. 
It  will  be  observed  that  this  coupling  admits  of  rotation  but  in  one  direction, 
the  one  tending  to  bring  the  ends  of  the  shafts  toward  each  other  ;  the  reverse 
motion  tends  to  unscrew  and  throw  them  apart,  arid  uncouple  them. 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


261 


FIG.  536. 


FIG.  537. 


Fig.  538  is  a  clamp  coupling  for  a  square  shaft. 

William   Sellers  &  Co.,  Philadelphia,  make  a  double-cone  vice  coupling, 
which  is  largely  used  (Fig.  539).     It  is  shown  complete  on  shaft  at  A.     B  is 
the   outer  shell   or  sleeve,    C   the   two 
<3ones,  and  D  the  bolts.     The  sleeve  is 
cylindrical   outside,    but   bored   with   a 
double  taper  inside,  smallest  at  center. 
The  cones  are  bored  to  fit  the  shaft,  and 
turned  outside  to  fit  the  interior  cones 
of  the  sleeve.      There  are  three  bolt- 
grooves  in  the  cones  and  sleeve,  and  one  is  cut  through  to  give  elasticity  to  the 
cones.     The  sleeve  and  cones  are  adjusted  over  the  joint  of  the  shafts,  leaving 
at  an  easy  fit  some  f "  between  the  ends  of  the  cones  ;  if  now  the  bolts  be  intro- 


FIG.  538. 


FIG.  539. 

duced  and  screwed  up,  the  cones  are  brought  nearer  to  each  other,  and  the 
shafts  are  securely  clamped  together.  Fig.  540  shows  the  coupling  in  section. 
In  many  cases  it  occurs  that  rigid  couplings,  such  as  we  have  given,  are 
objectionable  ;  they  necessarily  imply  that,  to  run  with  the  least  strain  possible, 
the  bearings  should  be  in  accurate  line  ;  any  displacement  involves  the  spring- 
ing of  the  shaft,  and,  if  considerably  moved,  fracture  of  shaft  or  coupling. 


FIG.  540. 


FIG.  541. 


Wherever,  then,  from  any  cause  the  alignment  can  not  be  very  nearly  accu- 
rate, some  coupling  that  admits  of  lateral  movement  should  be  adopted.  The 
simplest  of  these  is  the  box  or  sleeve  coupling  (Fig.  541),  sliding  over  the  end 
of  two  square  shafts,  keyed  to  neither,  but  often  held  in  place  by  a  pin  passing 
through  the  coupling  into  one  of  the  shafts.  For  round  shafts,  the  loose  sleeve 


262 


MACHINE  DESIGN  AND   MECHANICAL  CONSTRUCTIONS. 


coupling  is  a  pipe  or  hub,  generally  4  to  6  times  the  diameter  of  the  shaft  in 
length,  sliding  on  keys  fixed  on  either  shaft. 

Fig.  542  represents  a  horned  coupling.  The  two  parts  of  the  coupling  are 
counterparts  of  each  other,  each  firmly  keyed  to  its  respective  shaft,  but  not 
fastened  to  each  other  ;  the  horns  of  the  one  slip  into  the  spaces  of  the  other  ; 

if  the  faces  of  the  horns  are 
accurately  fitted,  it  affords 
an  excellent  coupling,  and 
is  not  perfectly  rigid. 

It  often  happens  that 
some  portion  of  a  shaft  or 
machine  is  required  to  be 
stopped  while  the  rest  of 
the  machinery  continues  in 
motion.  It  is  evident  that, 

if  one  half  of  a  horned  coupling  be  not  keyed  to  the  shaft,  but  permitted  to 
slide  lengthways  on  the  key — the  key  being  fixed  in  the  shaft,  forming  in  this 
case  what  is  more  usually  called  a  feather — by  sliding  back  the  half  till  the 
horns  are  entirely  out  of  the  spaces  of  the  other  half,  communication  of  motion 
will  cease  from  one  shaft  to  the  other. 


FIG.  542. 


FIG.  543. 

Fig.  543  represents  a  coupling  of  this  sort  for  a  large  shaft,  from  the  Corliss 
Steam-Engine  Company.  The  horns  are  8  in  number  on  each  part,  and  are 
thrown  readily  in  or  out  of  action  by  the  handle  h  turning  the  loose  part  of  the 
clutch  on  the  screw  cut  on  the  shafts. 

Fig.  544  is  another  form  of  disengaging  a  large  pulley  from  a  main  shaft, 
from  the  Corliss  Steam-Engine  Company.  The  pulley  is  fastened  to  a  cast- 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS.  263 


iron  pipe  or  sleeve  p  through  which  the  main  shaft  s  passes.  The  two  are 
attached  by  means  of  the  coupling  c,  one  half  of  which  is  attached  to  the  shaft 
and  the  other  to  the  sleeve.  When  bolted  together,  the  pulley  and  main  shaft 


264 


MACHINE   DESIGN   AND  MECHANICAL   CONSTRUCTIONS. 


move  together  ;  but  if  the  bolts  be  removed,  then  the  pulley  becomes  stationary 
even  if  the  shaft  is  running.  Shaft  and  sleeve  have  independent  bearings. 
A  (Fig.  544)  is  a  section  of  the  coupling  on  a  larger  scale,  and  shows  the  strong 
taper  of  the  bolts  without  head. 

Couplings  are  made  on  this  principle,  called  slide  or  clutch  couplings, 
when  the  motion  is  required  but  in  one  direction.  The  general  form  of  this 
coupling  is  given  in  Fig.  545.  A  represents  the  half  of  the  coupling  that  is 

keyed  to  the  shaft,  B  the 

6  ^.-^  sliding  half,  c  the  handle  or 

lever  which  communicates 
the  sliding  movement ;  the 
upper  end  of  the  lever  ter- 
minates, in  a  fork,  inclosing 
the  hub  of  the  coupling,  and 
fastened  by  two  bolts  or  pins 
to  a  collar  c'  round  the  neck 
of  the  hub  ;  1)  is  a  box  or 
bearing  for  the  shaft  A  ;  to 

support  B  the  end  of  its  shaft  extends  a  slight  dis- 
tance into  the  coupling  A.  Shafts  can  not  be  en- 
gaged with  this  form  of  coupling  while  the  driving 
shaft  is  in  motion,  without  great  shock  and  injury  to  the 
machinery.  To  obviate  this,  other  forms  of  coupling 
are  requisite  ;  one  of  these  is  represented  (Fig.  546). 
On  the  shaft  B  is  fixed  a  drum  or  pulley,  which  is 
embraced  by  a  friction  band  as  tightly  as  may  be 
found  necessary ;  this  band  consists  of  two  straps  of 
iron,  clamped  together  by  bolts,  leaving  ends  project- 
ing on  either  side  ;  the  portion  of  the  coupling  on  the  shaft  A  is  the  common 
form  of  bayonet  clutch  ;  the  part  c  c  is  fixed  to  the  shaft,  and  affords  a  guide 
to  the  prongs  or  bayonets  b  b,  as  they  slide  in  and  out.  Slipping  these  prongs 
forward,  they  are  thrown  into  gear  with 
the  ears  of  the  friction  band  ;  the  shaft 
A  being  in  motion,  the  band  slips  round 
on  its  pulley  till  the  friction  becomes 
equal  to  the  resistance,  and  the  pulley 
gradually  attains  the  motion  of  the 
clutch. 

But  of  all  slide  couplings,  to  engage 
and  disengage  with  the  least  shock  and 
at  any  speed,  the  friction  cone  coup- 
ling (Fig.  547)  is  by  far  the  best.  It 
consists  of  an  exterior  and  interior 
cone,  a,  b ;  a  is  fastened  to  the  shaft 

A,  while  b  slides  in  the  usual  way  on  the  feather  /  of  the  shaft  B  ;  pressing  b 
forward,  its  exterior  surface  is  brought  in  contact  with  the  interior  conical  sur- 
face of  a  ;  this  should  be  done  gradually  ;  the  surfaces  of  the  two  cones  slip  on 


FIG.  545. 


FIG.  546. 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


265 


each  other  till  the  friction  overcomes  the  resistance,  and  motion  is  transmitted 

comparatively  gradually,  and  without  danger  to  the  machinery.     The  longer 

the  taper  of  the  cones,  the  more 

difficult  the  disengagement ;   but 

the  more  blunt  the  cones,  the  more 

difficult  to  keep  the  surfaces  in 

contact.     An  angle  of  8°  with  the 

line  of  shaft  is  a  very  good  one 

for  surfaces  of  cones  of  cast-iron 

on  cast-iron.     When  thrown  into 

ifaas^^sai 

FIG.  547. 


FIG.  548. 

gear,  the  handle  of  the  lever  or  shipper  is  slipped  into  a  notch,  that  it  may  not 
be  thrown  out  by  accident. 

The  objection  to  this  coupling  is  that  it  will  work  out  of  gear  unless  the 
shipper-handle  is  held  firmly  in  its 
position,  and  producing  considerable 
friction  against  the  collar.  To  obvi- 
ate this  the  shipper  is  made  to  act  on 
a  toggle-joint  fastened  to  the  shaft, 
and,  once  thrown,  the  pressure  is 
self-continued,  and  preserved  with- 
out any  action  of  the  shipper,  and 
without  friction. 

Fig.  548  represents  a  double-fric- 
tion clutch,  of  the  Weston-Oapen 
patent.  The  clutch  G  is  slid  over 
the  toggle,  and  the  friction  cone  is 
forced  into  the  pulley  and  engaged 
therewith.  In  the  figure,  D'  is  thus 
engaged  with  A',  while  D  and  A  are 
not  in  contact. 

Fig.  549  is  a  perspective  of  the 
Mason  clutch,  in  which  two  toggles  FIG.  549. 

are  attached  to  the  sliding  hub  F. 

By  the  action  of  the  shipper  moving  the  hub  inward  the  two  toggles  force  the 
two  segments  E  E  against  the  inner  periphery  of  the  pulley,  which  is  turned 


266 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


parallel  with  the  axis.  The  toggles  are  so  adjusted  that  when  forced  in  they  are 
a  trifle  within  the  straight  line,  so  that  there  is  no  tendency  for  them  to  fly  out. 
Pulleys  are  used  for  the  transmission  of  motion  from  one  shaft  to  another 
by  the  means  of  belts  ;  by  them  every  change  of  velocity  may  be  effected.  The 
speed  of  two  shafts  will  be  to  each  other  in  the  inverse  ratio  of  the  diameter  of 
their  pulleys.  Thus,  if  the  driving  shaft  make  100  revolutions  per  minute, 
and  the  driving  pulley  be  18  inches  in  diameter,  while  the  driven  pulley  is  12 
inches,  then,  ^  .  lg  ..  100  .  15Q  . 

that  is,  the  driven  shaft  will  make  150  revolutions  per  minute.  Where  there 
is  a  succession  of  shafts  and  pulleys,  to  Jind  the  velocity  of  the  last  driven 
shaft :  Multiply  together  all  the  diameters  of  the  driving,  pulleys  by  the  speed 
of  the  first  shaft,  and  divide  the  product  by  the  product  of  the  diameters  of  all 
the  driven  pulleys. 


FTG.  550. 


FIG.  551. 


FIG.  552. 


Pulleys  are  made  of  cast-iron  and  of  every  diameter,  from  2  inches  up  to 
20  feet.  The  number  of  arms  vary  according  to  the  diameter ;  for  less  than 
8  inches  diameter  the  plate  pulley  is  preferable  (Fig.  550)  ;  that  is,  the  rim  is 
attached  to  the  hub  by  a  plate  ;  for  pulleys  of  larger  diameters,  those  with 
arms  are  used,  never  less  than  4  in  number.  The  arms  are  made  usually 
straight  (Fig.  551),  sometimes  curved  (Fig.  552). 


FIG. 


553. 


FIG.  554. 


MACHINE   DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


267 


Fig.  553  represents  a  portion  of  the  elevation  of  a  pulley  sufficient  to  show 
the  proportion  of  the  several  parts,  and  Fig.  554  a  section  of  the  same.  The 
parts  may  be  compared  proportionately  with  the  diameter  of  shaft ;  thus  the 
thickness  of  the  hub  is  about  -J-  the  diameter  of  the  shaft ;  this  proportion  is  also 


used  for  the  hubs  of  couplings  ;  the  width  of  the  arms  from  £  to  full  diame- 
ter ;  the  thickness  half  the  width  ;  the  thickness  of  the  rim  from  1  to  -J-  the 
diameter  ;  the  length  of  hub  the  same  as  the  width  of  face. 

Fig.  555  is  a  large  pulley  of  the  Southwark  Foundry  pattern.     The  hub 
is  cast  with  four  divisions,  to  admit  of  contraction  in  cooling,  and  the  rim  is  in 


268 


MACHINE   DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


halves,  to  admit  of  the  pulley  being  put  on  the  shaft  without  removing  it  from 
its  bearings.  This  is  now  very  common  practice  with  large  pulleys.  Wrought- 
iron  rim-pulleys  have  lately  been  introduced  in  which  the  spider — that  is,  the  hub 

and  arms — are  of  cast-iron,  and  a  wrought-iron  plate- 
rim  is  bolted  to  flanges  on  the  extremities  of  the  arms. 
Fig.  556  represents  a  faced  coupling  pulley,  an 
3  expedient  sometimes  adopted  when  a  joint  occurs 
where  a  pulley  is  also  required  ;  the  two  are  then 
combined ;  the  pulley  is  cast  in  halves — two  plate 
pulleys,  with  plates  at  the  side  instead  of  central, 
faced  and  bolted  together. 

Wooden  pulleys  are  commonly  called  drums  ;  these  are  now  but  seldom 
used  except  for  pulleys  of  very  wide  face.  Fig.  557  represents  one  form  of 
construction  in  elevation  and  longitudinal  section.  It  consists  of  two  cast-iron 
pulleys  A  A,  with  narrow  rims  ;  they  are  keyed  on  to  the  shaft  at  the  required 


FIG.  556. 


w 

I  - 


M^-i 


FIG.  557. 

distance  from  each  other,  and  plank  or  lagging  is  bolted  on  the  rims  to  form 
the  face  of  the  drum  ;  the  heads  of  the  bolts  are  sunk  beneath  the  surface  of 
the  lagging,  and  the  face  is  turned. 

Fig.  558  represents  a  wooden  pulley  which  may  be  termed  a  wooden  plate 

pulley.  The  plate  consists  of  sectors  of 
inch  boards  firmly  glued  and  nailed  to- 
gether, the  joints  of  the  boards  being 
always  broken.  The  face  is  then  formed 
in  a  similar  way,  by  nailing  and  gluing 
arcs  of  board  one  to  another  to  the  re- 
quired width  of  face  ;  these  last  should 
be  of  clear  stuff.  The  whole  is  retained 
on  the  shaft  by  an  iron  hub,  cast  with  a 
plate  on  one  side,  and  another  separate 
plate  sliding  on  to  the  hub  ;  the  hub  is 
placed  in  the.  center  of  the  pulley,  the 

two  plates  are  brought  in  contact  with  the  sides  of  the  pulley,  and  bolted 
through  ;  the  face  of  the  pulley  is  now  turned  in  the  lathe.  A  similar  arrange- 
ment of  hub  is  used  for  the  hanging  of  grindstones. 


FIG.  558. 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS.  269 

V^> 

Cone  pulleys  are  used  to  change  the  speed  of  the  driven  shaft.     Fig.  559 

represents  a  cone  pulley  c  on  a  shaft  with  a  fast  pulley  d,  or  one  attached  to  the 
shaft,  while  the  other,  e,  is  loose  and  revolves  on  it.  The  cone  pulleys  are  for 
changes  of  speed  on  the  machine,  which  has  upon  it  another  set  of  cone  pul- 
leys, but  in  reverse  position,  the  small  one  being  opposite  the  large  one  on  the 


FIG.  559. 

counter  shaft.  A  counter  shaft  is  one  disconnected  from  a  main  or  leading 
shaft,  for  the  purpose  of  driving  a  machine.  This  counter  is  connected  with 
the  main  shaft  by  a  belt  from  a  pulley  on  this  shaft  passing  over  the  fixed  or 
loose  pulley.  When  on  the  fixed  pulley,  the  counter  shaft  is  moved ;  when  on 
the  loose  pulley  it  revolves  on  the  shaft,  and  the  shaft  is  still.  To  move  the 
belt,  there  is  a  fork  between  which  the  sides  of  the  belt  approaching  the  coun- 
ter passes,  and  a  movement  of  this  fork  by  a  shipper  throws  the  counter  in  or 
out  of  movement.  The  faces  of  the  fast  or  loose  pulleys  are  made  flat,  and 
provision  is  to  be  made  for  oiling  the  inside  of  the  hub  of  the  loose  pulley, 
which  is  done  by  oil-holes  and  grooves. 


FIG.  560. 

It  is  often  necessary  to  reverse  the  motion  of  a  machine.  This  is  readily 
done  by  a  system  of  fast  and  loose  pulleys,  as  shown  in  the  plan  and  elevation, 
Fig.  560,  in  which  A  is  a  drum  or  wide-faced  pulley  on  the  driving-shaft,  B  a 
fast  pulley  on  the  driven  shaft,  and  C  and  D  loose  pulleys  on  the  same.  The 
action  will  be  understood  from  the  direction  of  the  arrows.  The  driving-shaft 


270 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


revolves  always  in  the  same  direction,  but  on  the  driven  shaft  the  loose  pulley 
of  the  straight  belt  is  drawn  from  the  bottom,  and  partakes  of  the  same  motion 
as  the  driving-pulley  ;  while  by  the  cross-belt  the  draft  is  at  the  top  of  its  pul- 
ley, and  the  motion  reversed.  If  the  straight  or  open  belt  be  shipped  on  to  the 
fast  pulley  B,  the  motion  given  to  the  shaft  is  like  that  of  the  driving-shaft ;  if 
the  cross-belt  be  shipped  on  to  the  fast  pulley,  the  motion  of  the  shaft  is  re- 
versed. It  will  be  observed  in  the  elevation,  the  lower  side  of  the  open  belt  is 
straight,  while  there  is  a  sag  in  the  upper  ;  the  first  is  called  the  tight  or  lead- 
ing belt,  it  being  the  belt  through  which  the  power  is  transmitted,  while  the 
upper  side  is  the  loose  or  slack  belt.  The  stress  on  the  tight  belt  is  equal  to 
that  of  the  power  transmitted,  and  the  stress  with  which  the  belt  is  stretched 
over  the  pulleys,  so  that  it  will  not  slip  in  conveying  this  power. 

When  the  belt  is  shifted,  while  in  motion,  to  a  new  position  on  a  drum  or 
pulley,  or  from  fast  to  loose  pulley,  or  vice  versa,  the  lateral  pressure  must  be 
applied  on  the  advancing  side  of  the  belt,  on  the  side  on  which  the  belt  is  ap- 
proaching the  pulley,  and  not  on  the  side  on  which  it  is  running  off.  It  is  only 
necessary  that  a  belt,  to  maintain  its  position,  should  have  its  advancing  side 
in  the  plane  of  rotation  of  that  section  of  the  pulley  on  which  it  is  required  to 
remain,  without  regard  to  the  retiring  side.  On  this  principle,  motion  may  be 
conveyed  by  belts  to  shafts  at  any  angle  to  each  other.  Let  A  and  B  (Fig. 
561)  be  two  shafts  at  right  angles  to  each  other,  A  vertical,  B  horizontal,  so 
that  the  line  run  perpendicular  to  the  direction  of  one  axis  is  also  perpendicu- 
lar to  the  other,  and  let  it  be  required  to  connect  them  by  pulleys  and  a  belt, 


FIG.  561. 


FIG.  562. 


that  their  direction  of  motion  may  be  as  shown  by  the  arrows  and  their  veloci- 
ties as  3  of  A  to  2  of  B.  On  A  describe  the  circumference  of  the  pulley  pro- 
posed on  that  shaft  ;  to  this  circumference  draw  a  tangent  a  b  parallel  to  m  n  ; 
this  line  will  be  the  projection  of  the  edge  of  the  belt  as  it  leaves  A,  and  the 
center  of  the  belt  as  it  approaches  B  ;  consequently,  lay  off  the  pulley  b  on  each 
side  of  this  line,  and  of  a  diameter  proportional  to  the  velocity  required.  To 
fix  the  position  of  the  pulley  on  A,  let  Fig.  562  be  another  view  taken  at  right 
angles  to  Fig.  561,  and  let  the  axis  B  have  the  direction  of  motion  indicated  by 
the  arrow,  then,  the  circle  of  the  pulley  being  described,  and  a  tangent  a'  V 
drawn  to  it  perpendicular  to  the  axis  B  as  before  determined,  the  position  of 
the  pulley  on  the  shaft  A  is  likewise  fixed. 

The  positions  of  the  two  pulleys  are  thus  fixed  in  such  a  way  that  the  belt 
is  always  delivered  by  the  pulley  it  is  receding  from  into  the  plane  of  rotation 
of  the  pulley  toward  which  it  is  approaching.  If  the  motion  be  reversed,  the 
belt  will  run  off. 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


271 


Figs.  563  and  564  are  the  plan  and  elevation,  on  a  large  scale,  of  a  similar 
arrangement  of  pulleys  and  belts. 

It  is  not  an  essential  condition  that  the  shafts  should  be  at  right  angles 
to  each  other  to  have  motion  transferred  by  a  belt.  They  may  be  placed  at 


Fm.  564. 

any  angle  to  each  other,  provided  the  shafts  lie  in  parallel  planes,  so  that 
the  perpendicular  drawn  to  one  axis  is  perpendicular  to  the  other.  If  other- 
wise, recourse  must  be  had  to  guide-pulleys,  which  should  be  considerably 

convex  on  their  face. 

Fig.  565  is  an  arrangement  adopted  in  port- 


FIG.  566. 


FIG.  565. 


FIG.  567. 


272 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


able  grist-mills  for  driving  the  vertical  shafts  «,  I,  of  mill-stones,  from  pulleys 
on  a  horizontal  shaft.     Here  it  is  thought  necessary  to  use  guide-pulleys. 

Figs.  566  and  567  are  the  elevation  and  plan  of  another  arrangement  of 
pulleys  and  guide-pulleys  ;  a  ~b  is  the  intersection  of  the  middle  plane  of  the 
principal  pulleys.     Select  any  two  points  a  and  b  on  this  line,  and  draw  tan- 
gents a  c,  b  d,  to  the  principal  pulleys.     Then  c  a  c  and  a  b  d  are  suitable  direc- 
tions  for   the    belt.      The    guide-pulleys   must  be 
placed  with  their  middle  planes  coinciding  with  the 
planes  c  a  c  and  a  b  d.     The  belt  will  run  in  either 
direction. 

Fig.  568  is  a  perspective  of  a  hanger  of  William 
Sellers  &  Co.,  in  which  the  guide-pulleys  can  be  ad- 
justed to  revolve  in  the  required  plane. 

It  has  been  said  that  it  is  necessary  to  stretch  the 
belt  over  the  pulleys,  so  that  it  will  not  slip  while  con- 
veying the  power.  If  the  pulleys  are  horizontal,  the 
weight  of  the  belt  itself  may  provide  for  this  friction, 
and  this  friction  diminishes  with  the  inclination  of 
the  belt  till  it  becomes  vertical,  when  the  friction  of 
the  stretch  is  the  only  factor  of 
the  adhesion  of  the  belt  to  the 
lower  pulley  ;  and,  as  the  belt 
lengthens  by  use,  the  value  of 
this  friction  becomes  nothing. 
This  position  of  pulleys  should 
not  obtain  if  it  can  be  avoided ; 
but  if  not,  the  friction-stress 

should  be  by  means  of  a  binder  on  the  loose  belt.  The 
binder  (Fig.  569)  hangs  in  a  loose  frame  or  links,  and 
rests  on  the  belt,  so  that  the  weight  of  the  binder  and 
frame  tends  to  take  up  the  slack  of  the  belt.  Sometimes 
the  binder  is  forced  against  the  belt  by  a  screw  acting  on 
its  frame.  By  the  relief  of  the  binder  the  belt  becomes 
slack,  and  the  friction  of  the  belt  on  the  pulleys  may  be- 
come nothing,  and  motion  stopped.  On  many  machines 
and  lines  of  shafting  this  arrangement  for  engaging  and 
disengaging  is  made  use  of.  Binders  are  a  necessity  where 
the  two  pulleys  are  near  to  each  other,  either  to  increase 
the  bearing  surface  of  the  belt  on  the  pulleys  or  to  make 
up  for  the  slight  weight  of  a  short  belt.  Belts  run  the 
best  when  their  length  and  position  are  such  as  to  give  the 
frictional  stress  without  much  stretching  on  the  pulleys, 
and  without  binders.  It  is  also  necessary  that  the  surface  of  belt  in  contact 
with  the  pulleys  should  be  large,  as  the  frictional  stress  varies  with  the  surface. 
The  widths  of  belt  hereafter  given  are  based  on  the  usual  surface  of  about  180°, 
or  half  the  circumference  of  the  pulleys.  On  account  of  the  friction  and  wear 
it  is  usual  to  put  the  hair  side  of  the  belt  next  the  pulley. 


FIG.  568. 


FIG.  569. 


MACHINE  DESIGN   AND   MECHANICAL  CONSTRUCTIONS. 


273 


In  determining  the  necessary  length  for  any  position,  the  simplest  way  is  to 
measure  it,  if  the  construction  is  complete  ;  if  not,  to  make  a  drawing  of  the 
pulleys  in  position  to  a  scale,  and  measure  on  the  drawing. 

The  width  of  the  belts  should  always  be  a  little  less  than  the  face  of  the 
pulley  ;  both  are  to  be  determined  by  the  power  to  be  transmitted  and  the 
velocity  of  movement.  For  the  lighter  stress  of  belt  a  single  thickness  is  only 
necessary,  but  for  belts  from  prime  movers,  transmitting  great  power,  double 
belts  are  used. 

For  single  belts,  embracing  180°  of  the  circumference,  with  a  velocity  of  10 
feet  per  second,  one  horse-power  can  be  transmitted  for  each  inch  in  width  of 
belt,  with  a  maximum  stress  on  the  belt  of  50  pounds,  and  pressure  on  journals 
of  about  85  pounds  per  inch  of  width  of  belt. 

D  X  TT  X  R 

John   T.    Henthorn's    formula    for  double    belts 


is 


450 


=  H.  P. 


per  inch  in  width,  in  which  D  is  the  diameter  of  pulley  in  feet,  E  the  revolu- 
tions per  minute.     This  is  expressed  graphically  in  Fig.  570. 

Diameter  of  Pulleys,  shown  by  Diagonals. 


Horse-Power  per  Inch  of  Width. 
FIG.  570. 

Use  of  Diagram. — To  find  the  horse-power  that  can  be  transmitted  by  a 
24"  belt  on  a  20-foot  pulley  making  100  revolutions  per  minute  :  The  abscissa 
line  100  intersects  the  diagonal  20  on  the  ordinate  line  14  ;  14  X  24  =  336  = 
horse-power  transmissible. 

To  find  the  belt  necessary  to  transmit  100  horse-power  through  a  10-foot 
pulley  and  120  revolutions  per  minute  of  shaft :  The  abscissa  120  cuts  the 

diagonal  10  on  the  ordinate  line  8£;  -  -  =  12"  width  of  belt.     If  the  pulley 
18  8£ 


274  MACHINE  DESIGN   AND  MECHANICAL  CONSTRUCTIONS. 

were  12-foot  instead  of  10,  it  will  be  seen  by  the  diagram,  the  intersection  of 

100 
diagonal  would  be  at  10,  and  the  width  of  belt  —  =  10". 

The  above  rules  are  applicable  to  leather  belts,  but  belts  of  India-rubber 
and  canvas  are  largely  used,  and  can  be  procured  of  any  desirable  dimensions, 
and  the  strength  and  adhesion  are  generally  considered  greater  than  those  of 
leather  belts.  They  are  especially  valuable  in  situations  exposed  to  wet,  where 
leather  is  not  admissible. 

It  is  the  present  practice  to  run  belts  at  high  speed  ;  5,000  to  6,000  feet  is 
admissible  with  suitable  pulleys  and  position. 

The  use  of  ropes  instead  of  belts  has  not  obtained  largely  in  this  country, 
but  in  a  late  report  of  Mr.  Edward  Atkinson  on  English  practice,  he  says 
that  in  first-class  mills  ropes  instead  of  leather  belts  have  taken  the  place  of 
the  upright  shafts  and  gears  which  were  formerly  used.  He  instances  in 
one  mill, 

"  The  main  wheel  on  the  2d-motion  shaft  from  the  engine  or  driving-pulley 
is  grooved  ;  it  is  12  feet  in  diameter,  104  revolutions  per  minute,  and  has  20 
ropes. " 

"  The  rules  for  rope-driving  have  been  given  me  as  follows  : 

"1.  Never  use  pulleys  of  less  diameter  than  six  feet  for  main  work. 

<(  2.  The  greater  the  velocity  of  the  rope  per  minute  the  greater  the  effi- 
ciency, up  to  5,000  feet  per  minute. 

"3.  For  great  power,  ropes  Scinches  diameter,  2  inches  when  stretched, 
are  best  ;  cable-laid  with  3  strands,  and  each  strand  of  3  finer  strands.  Where 
small  power  is  required  it  is  not  necessary  to  have  the  rope  cable-laid,  or  so 
great  in  diameter.  For  ropes  of  small  diameter  smaller  diameters  of  pulleys 
may  be  used  than  6  feet,  and  cotton  ropes  are  preferable  to  hemp.  For  large 
ropes  or  outside  work,  hemp  is  better  than  cotton.  Cotton  ropes  made  from 
yarn,  counts  about  20  to  30,  are  better  than  those  made  from  rovings. 

"  4.  With  a  rope  2%  inches  diameter,  and  pulleys  above  6  feet,  each  rope 
will  drive  10  indicated  horse-power  every  1,000  feet  of  rope-speed  per  minute. 

"  5.  Whenever  circumstances  will  allow,  the  slack  side  of  the  ropes  ought 
always  to  be  on  the  top,  so  as  to  keep  the  rope  tight  in  the  groove  where  it 
stretches. 

1  'The  tarred  cotton  rope  and  tarred  spindle  banding,  thoroughly  impreg- 
nated with  pine  tar,  are  reasonably  supple,  perfectly  free  from  stickiness,  and 
are  said  to  be  very  non-elastic  and  substantially  free  from 
the  effects  of  humidity." 

Fig.  571  represents  a  cross-section  of  single-grooved  rim 
for  a  cotton  or  hemp  rope  as  used  in  this  country,  the 
groove  being  simply  turned  and  polished. 

Fig.  572  is  a  cross-section  of  the  rim  of  wheel  for  wire 
FIG.  571.     FIG.  572.      rope,  showing  the  rubber  lining  contained  in  a  dovetailed 
recess  at  the  bottom  of  the  groove. 

From  the  circular  of  Messrs.  Roebling  &  Sons  we  make  the  following 
table  of  transmission  of  power  by  wire,  the  number  of  revolutions  per  minute 
being  100  : 


MACHINE  DESIGN   AND  MECHANICAL  CONSTRUCTIONS. 


275 


Diameter  of 
wheel  in 
feet. 

Diameter  of  rope. 

Horse-power. 

Diameter  of 
wheel  in 
feet. 

Diameter  of  rope. 

Horse-power. 

4 

f 

3-3 

10 

ttt 

68-7 

/TO. 

4 

f 

4-1 

5 

A 

8-6 

11 

f  « 

81-1 
94-4 

6 

7 

* 

Ia6 

13-4 
21-1 

12 

Ht 

116-7 
124-1 

8 

i 

27-5 

13 

HI 

140- 
153-2 

9 

1%    1 

50- 
51-9 

14 

f  * 

185' 
176- 

10 

IH 

55- 

58-4 

15 

*  * 

259' 
259- 

Gearing. — The  term  gearing,  in  general  sense,  is  applied  to  all  arrange- 
ments for  the  transmission  of  power ;  it  is  also  used  in  a  particular  sense,  as 
toothed  gearing. 

Toothed  gearing  may  be  divided  into  two  great  classes — spur  and  level 
wheels.  In  the  former,  the  axes  of  the  driving  and  driven  wheels  are  parallel 
to  each  other  ;  in  the  latter  they  may  be  situated  at  any  angle  ;  if  of  equal  size 
and  at  right  angles,  they  are  called  miter-gears. 

Spur-wheels,  strictly  so  called,  consist  of  wheels  of  which  the  teeth  are  dis- 
posed at  the  outer  periphery  of  the  wheel  (Fig.  583),  in  direction  of  radii  from 
their  centers. 

Internal  gearing,  in  which  the  teeth  are  disposed  in  the  interior  periphery 
of  the  wheel,  in  direction  of  radii  from  their  centers  (Fig.  596). 

Rack-gear  and  pinion  are  employed  to 
convert  a  rotatory  into  a  rectilinear  mo-   T' 
tion,  or  vice  versa.     In  this  arrangement 
the  pinion  is  a  spur-wheel,  acting  on  teeth 
placed  along  a  straight  bar  (Fig.  595). 


FIG.  573. 


Fm.  574. 


Bevel-gearing,  strictly  so  called,  consists  of  toothed  wheels  formed  to  work 
together  in  different  planes,  their  teeth  being  disposed  at  an  angle  to  the  plane 
of  their  faces  (Fig.  591). 


276 


MACHINE  DESIGN  AND   MECHANICAL  CONSTRUCTIONS. 


Trundle-pins  or  wheels  (Fig.  578)  are  constructed  with  cylindrical  pieces,, 
called  staves  or  pins,  instead  of  teeth.  Fig.  573  is  an  illustration  of  trundle- 
gears  with  wooden  pins  ;  the  pinion  with  double  plates  is  called  a  lantern. 
This  construction  is  very  useful  when  iron  gears  can  not  be  easily  got  or  re- 
paired. The  trundle  may  be  used  either  with  a  spur-wheel  to  transmit  motion 
to  parallel  shafts,  or  with /ace  or  crown  wheels. 

The  primary  object  of  toothed  gears  is  the  uniform  transmission  of  power 
supposed  to  be  constant  and  equal  ;  the  one  wheel  conducts  the  other,  and 
they  are  designated  severally  as  driver  and  driven,  or  leader  and.  follower.  There 
must  be  a  central  line  of  contact  of  the  teeth,  when  the  surfaces  move  with  the 
same  velocity.  In  spur-wheels  this  line  of  contact  is  represented  by  circles,  as 
A  and  B  (Fig.  574).  These  circles  are  called  pitch-circles — they  must  have 
the  same  angular  velocity,  and  the  number  of  revolutions  of  each  wheel  in  a 
given  time  must  be  inversely  as  their  diameters. 

To  find  the  relative  radii  of  two  wheels  whose  number  of  revolutions  are 
known  :  Divide  the  distances  between  their  centers  into  parts  inversely  pro- 
portional to  the  number  of  revolutions  which  the  wheels  are  to  make  in  the 
same  unit  of  time.  Thus,  let  A  and  B  (Fig.  574)  be  the  given  centers,  the 
ratio  of  their  velocities  being  respectively  two  and  three  ;  if  the  line  joining  the 
centers  A  and  B  be  divided  into  2  -f  3  =  5  equal  parts,  that  is,  into  as  many 
equal  parts  as  there  are  units  in  the  terms  of  the  given  ratio,  the  radius  of  the 
wheel  upon  A  will  contain  three  of  these  parts,  and  the  radius  of  the  pinion  on 
B  will  contain  the  remaining  two  parts. 

The  sizes  of  a  pair  of  bevels  are,  however,  limited  to  no  particular  diame- 
ters as  when  the  axes  are  parallel ;  the  wheels  may  be  made  of  any  convenient 
sizes,  and  the  teeth  consequently  of  any  breadth,  according  to  the  stress  they 
are  intended  to  bear.  The  question  is  the  mode  of  determining  the  relative 
sizes  of  the  pair  ;  and  this  resolves  itself  into  a  division  of  the  angle  included 

between  the  two  axes  inversely  as 
the  ratio  of  their  angular  velojci- 
ties.  Let  B  and  C  (Fig.  575)  be 
the  position  of  the  two  given  axes, 
and  let  them  be  prolonged  till 
they  meet  in  a  point  A.  Further, 
let  it  be  required  that  C  make 
seven  revolutions  while  B  makes 
four.  From  any  points  D  and  E 
in  the  lines  A  B,  AC,  and  per- 
pendicular to  them,  draw  D  cl  and 
E  e  of  lengths  (from  a  scale  of 
equal  parts)  inversely  as  the  num- 
ber of  revolutions  which  the  axes 
are  severally  required  to  make  in 

the  same  unit  of  time.  Thus,  the  angular  velocity  of  axis  B  being  4  (Fig. 
575),  and  that  of  the  axis  C  being  7,  the  line  D  d  must  be  drawn  =  7,  and  the 
line  E  e  =  4.  Then  through  d  and  e  parallel  with  the  axes  A  B  and  A  C  draw 
d  c  and  e  c  till  they  meet  in  c.  A  straight  line  drawn  from  A  through  c  will 


Fm.  575. 


MACHINE  DESIGN  AND   MECHANICAL  CONSTRUCTIONS. 


277 


then  make  the  required  division  of  the  angle  BAG,  and  define  the  line  of 
contact  of  the  two  cones,  by  means  of  which  the  two  rolling  frusta  may  be  pro- 
jected at  any  convenient  distance  from  A. 

Otherwise,  having  determined  the  relative  perimeters,  diameters,  or  radii,  of 
the  pair,  then  the  lines  D  d  and  E  e  are  to  each  other  directly  as  these  quanti- 
ties. B  F  and  C  F  are  radii  of  the  pitch-circle. 

The  case  in  which  the  axes  are  neither  parallel  nor  intersecting  admits  of 
solution  by  means  of  a  pair  of  bevels  upon  an  intermediate  axis,  so  situated  as 
to  meet  the  others  in  any  convenient  points. 

When  the  contiguity  of  the  shafts  is  such  as  to  permit  of  their  being  con- 
nected by  a  single  pair,  skewed  bevels  are  sometimes  employed. 

When  the  axes  are  at  right  angles  to  each  other,  and  do  not  intersect,  the 
wheel  and  screw  may  be  employed  to  connect  them.  The  velocity  of  motion  is 
in  this  arrangement  immediately  deduced  from  that  of  the  screw,  its  number 
of  threads,  and  the  number  of  teeth  in  its  gearing- wheel.  Thus,  if  it  be  required 
to  transmit  the  motion  of  one  shaft  to  another,  contiguous  and  at  right  angles 


100          1,000          2,000          3,000          4,000          5,000          6,000 

Stress  in  Pounds. 
FIG.  576. 


7,000          8,000      9,000 


to  it — the  angular  motions  being  as  20  to  1 — then,  if  the  screw  be  a  single- 
threaded  one,  the  wheel  must  have  20  teeth  ;  but  if  double-threaded,  the  num- 
ber of  teeth  will  be  increased  to  40,  for  2  teeth  will  be  passed  at  every  revolu- 
tion. If  the  screw  have  few  threads  compared  with  the  number  of  teeth  of  the 
wheel,  it  must  always  assume  the  position  of  driver  on  account  of  the  obliquity 
of  the  thread  to  the  axis  ;  and  in  this  respect  its  action  is  analogous  to  that  of 
a  traveling  rack,  moving  endwise  one  tooth,  while  the  screw  makes  one  revo- 
lution on  its  axis. 

If  the  pitch-circle  be  divided  into  as  many  equal  parts  as  there  are  teeth  to 
be  given  to  the  wheel,  the  length  of  one  of  these  parts  is  termed  the  pitch  of 


278 


MACHINE   DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


the  teeth.    One  of  these  arcs  comprehends  a  complete  tooth  and  space,  meaning 
by  space  the  hollow  opening  between  two  contiguous  teeth. 

The  pitch  depends  on  the  power  to  be  transmitted  or  the  stress  on  each 
tooth.  The  diagram  (Fig.  576)  is  by  John  T.  Henthorn,  M.  E.,  in  which 
pitch  and  face,  represented  by  multiples  of  the  pitch,  are  proportioned  to  the 
stress  in  pounds. 

If  the  pitch  be  known,  the  number  of  teeth  in  a  wheel  can  be  determined 
approximately  by  dividing  the  circumference  of  the  wheel  by  the  pitch,  but 
there  must  be  no  remainder  in  the  quotient — there  can  be  no  fraction  of  a 
pitch — either  the  pitch  or  diameter  of  wheel  must  be  changed  if  necessary  to 
produce  this  result ;  generally  the  latter,  as  gears  are  usually  made  of  determi- 
nate inches  and  fractions,  as  given  in  the  table,  by  which  also  calculation  for 
diameters  and  number  of  teeth  is  much  simplified. 

Example  1.— Given  a  wheel  of  88 
teeth,  2i-inch  pitch,  to  find  the  di- 
ameter of  the  pitch-circle.  Here  the 
tabular  number  in  the  second  column 
answering  to  the  given  pitch  is  '7958, 
which  multiplied  by  88  gives  70 -OS 
for  the  diameter  required. 

2.  Given  a  wheel  33  inches  diam- 
eter, If -inch  pitch,  to  find  the  num- 
ber of  teeth.  The  corresponding  fac- 
tor is  1*7952,  which,  multiplied  by 
33,  gives  59-242  for  the  number  of 
teeth — that  is,  59£  teeth  nearly.  Now 
59  would  here  be  the  nearest  whole 
number,  but  as  a  wheel  of  60  teeth 
may  be  preferred  for  convenience  of 
calculation  of  speeds,  we  may  adopt 
that  number,  and  find  the  diameter 
corresponding.  The  factor  in  the 
second  column  answering  to  If  pitch 
is  -557,  and  this  multiplied  by  Go 
gives  33'4  inches  as  the  diameter 
which  the  wheel  ought  to  have. 

Another  mode  of  sizing  wheels  in 
relation  to  their  pitches,  diameters, 
and  number  of  teeth,  is  adopted  in 
some  machine  shops,  by  dividing  the 
diameter  of  the  pitch-circle  into  as 
many  equal  parts  as  there  are  teeth 
to  be  given  to  the  wheel.  To  illus- 
trate this  by  an  arithmetical  example, 
let  it  be  assumed  that  a  wheel  of  20 
inches  diameter  is  required  to  have  40 
teeth  ;  then  the  diametral  pitch, 


p 

7T 

~     IT 

N  =     p     x  D. 

PITCH  IN 

HULK.—  To  find  the 

KTTLE.—  To  find  the 

INCHES 
AND 

diameter  in    inches, 

number    of   teeth, 

PARTS  OF 

multiply  the  number  i  multiply    the    given 

AN  INCH. 

of  teeth  by  the  tabu-    diameter   in    inches 

lar  number  answer-    by  the  tabular  mim- 

ing    to     the     given    ber  answering  to  the 

pitch. 

given  pitch. 

Values  of 
P. 

Values  of  -jp 

Values  of-p- 

6 

1-9095 

•5236 

5 

1-5915 

•6283 

4| 

1-4270 

•6981 

4 

1-2732 

•7854 

3£ 

1-1141 

•8976 

3 

•9547 

1-0472 

2f 

•8754 

1-1333 

a* 

•7958 

1-2566 

a* 

•7135 

1-3963 

2 

•6366 

1-5708 

1* 

•5937 

1-6755 

If 

•5570 

1-7952 

If 

•5141 

1-9264 

H 

•4774 

2-0944 

1$ 

•4377 

2-2848 

li 

•3979 

2-5132 

H 

•3568 

2-7926 

1 

•&18S 

3-1416 

i 

•2785 

3  '  5904 

t 

•2387 

4-1888 

I 

•1989 

5-0266 

I 

•1592 

6-2832 

1 

•1194 

8-3776 

i 

•0796 

12-5664 

MACHINE  DESIGN   AND  MECHANICAL   CONSTRUCTIONS. 


279 


20         l 

-  —  —  = 

40      m 

that  is,  the  diameter  being  divided  into  equal  parts  corresponding  in  number 
ttf  the  number  of  teeth  in  the  circumference  of  the  wheel,  the  length  of  each 
of  these  parts  is  |  an  inch,  consequently  m  =  2  ;  and  according  to  the  phrase- 
ology of  the  workshop,  the  wheel  is  said  to  be  one  of  two  pitch. 

In  this  mode  of  sizing  wheels,  a  few  determined  values  are  given  to  m,  as 
20,  16,  14,  12,  10,  9,  8,  7,  6,  5,  4,  3,  2,  1,  which  includes  a  variety  of  pitches 
from  -J-inch  up  to  3  inches,  according  to  the  following  table,  which  shows  the 
value  of  the  circular  pitches  corresponding  to  the  assigned  values  of  m. 


VALUES  OF  m. 

1. 

2. 

8. 

4. 

5. 

fi. 

7. 

8. 

9. 

10. 

12. 

14. 

16. 

20. 

Corresponding    eir-  } 
cular  pitch  in  dec-  > 

3-142 

1-571 

1-047 

•785 

•628 

•524 

•449 

•393 

•349 

•314 

•262 

•224 

•196 

•157 

imals  of  an  inch.  ) 

Fundamental  principle. — In  order  that  two  circles  A  and  B  (Fig.  577)  may 
be  made  to  revolve  by  the  contact  of  the  surfaces  of  the  curves  m  m  and  n  n  of 
their  teeth  precisely  as  they  would  by  the  friction  of  their  circumferences,  it  is 
necessary  and  sufficient  that  a  line  drawn  from  the  point  of  contact  t  of  the 
teeth  to  the  point  of  contact  c  of  the  circumferences  (pitch-circles)  should,  in 
every  position  of  the  point  t,  be  perpendicular  to  the  surfaces  of  contact  at  that 
point ;  that  is,  in  the  language  of  mathematicians,  that  the  straight  line  be  a 


QT& 


normal  to  both  the  curves  m  m  and  n  n.  The  principle  here  announced  ex- 
hibits a  special  application  of  one  particular  property  of  that  curve  known  to 
mathematicians  as  the  epicycloid  (see  page  30). 

Of  epicycloidal  teeth. — The  simplest  illustration  of  the  action  of  epicycloidal 
teeth  is  when  they  are  employed  to  drive  a  trundle,  as  represented  in  Fig.  578. 


280 


MACHINE   DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


Let  it  be  assumed  that  the  staves  of  the  trundle  have  no  sensible  thickness  ; 
that  the  distance  of  their  centers  apart,  that  is  their  pitch,  and  also  their  dis- 
tance from  the  center  of  the  trundle,  that  is  their  pitch-circle,  are  known. 
The  pitch-circles  of  the  trundle  and  wheel  being  then  drawn  from  their  respec- 
tive centers  B  and  A,  set  off  the  pitches  upon  these  circumferences,  correspond- 
ing to  the  number  of  teeth  in  the  wheel  and  number  of  staves  in  the  trundle  ; 
let  five  pins,  ale,  etc. ,  be  fixed  into  the  pitch-circle  of  the  trundle  to  represent 
the  staves,  and  let  a  series  of  epicycloidal  arcs  be  traced  with  a  describing  cir- 
cle, equal  in  diameter  to  the  radius  of  the  pitch-circle  of  the  trundle,  and  meeting 
in  the  points  Iclm  n,  etc.,  alternately  from  right  and  left.  If,  now,  motion  be 
given  to  the  wheel  in  the  direction  of  the  arrow,  then  the  curved  face  m  r  will 
press  against  the  pin  #,  and  move  it  in  the  same  direction  ;  but  as  the  motion 
continues,  the  pin  will  slide  upward  till  it  reaches  m,  when  the  tooth  and  pin 
will  quit  contact.  Before  this  happens,  the  next  pin  a  will  have  come  into 
contact  with  the  face  a  I  of  the  next  tooth,  which  repeating  the  same  action, 
will  bring  the  succeeding  pair  into  contact ;  and  so  on  continually. 

To  allow  of  the  required  thickness  of  staves,  it  is  sufficient  to  diminish  the 
size  of  the  teeth  of  the  wheel  by  a  quantity  equal  to  the  radius  of  the  staves 
(sometimes  increased  by  a  certain  fraction  of  the  pitch  for  clearance),  by  draw- 
ing within  the  primary  epicycloids,  at  the  required  distance,  another  series  of 
curves  parallel  to  these.  In  practice,  a  portion  must  be  cut  from  the  points  of 
the  teeth,  and  also  a  space  must  be  cut  out  within  the  pitch-circle  of  the  dri- 
ver, to  allow  the  staves  to  pass ;  but  no  particular  form  is  requisite,  the  con- 
dition to  be  attended  to  is  simply  to  allow  of  sufficient  space  for  the  staves  to 
pass  without  contact. 

It  is  a  common  practice  of  shops  to  take  as  the  diameter  of  the  rolling  circle 
the  radius  of  the  smallest  pinion  which  will  ever  be  used  for  gears  of  this  pitch, 
and  constructing  the  epicycloids  for  different  diameters  of  this  pitch,  and  allow- 
ing arcs  of  circles  corresponding  very  closely  to  these  epicycloids.  On  this 
principle,  Robert  Adcock,  0.  E.,  constructed  a  table  of  radii  for  these  arcs,  for 
rolling  circles  of  pinions  of  8,  10,  and  12  teeth.  We  give  the  last  only  as  an- 
swering the  conditions  of  practice  : 


•3 

® 

SMALLEST   PINION, 

•3 

e 

SMALLEST  PINION, 

1  5 

JJ 

SMALLEST   PINION, 

1 

*i 

TWELVE  TEETH. 

3 

ll 

TWELVR   TEETH. 

3 

*! 

TWELVE   TEETH. 

^ 

ii 

Eadiiofthe 

Eadii  of  the 

* 

s  -a 

Radii  of  the 

Radii  of  the 

1    o 

o 

:3  -a 

Radii  of  the 

Radii  of  the 

X 

•5  " 

facee 

of 

flanks  of 

3 

!* 

faces  of 

flanks  of 

•-SB 

faces  of 

flanks  of 

a 

w'5* 

teeth. 

teeth. 

£ 

M* 

teeth. 

teeth. 

1 

«'a 

teeth. 

teeth. 

12 

1-93 

1-880-75 

27 

4-31 

•23 

•84 

4-68 

•41 

42 

6'69 

6-601   -89 

6-91 

•20 

13 

2-09 

2-04 

0-76 

7-45 

7-14 

28 

•46 

•39 

•85 

•37 

•38 

43 

•85 

•76    '89 

7-06 

•20 

14 

2-25 

•19 

•77 

4-86 

4-27 

29 

•62 

•55 

•85 

•92 

•36 

44 

7-01 

92 

•89 

•22 

•19 

15 

2-40 

•35 

•78 

3-92 

3-04 

30 

•78 

•70 

•86 

5-07 

•34 

45 

•17 

7-07 

•89 

•38 

•18 

16 

2-56 

•50 

•78 

•62 

3-53 

31 

•94 

•86 

•86 

•21 

•32 

46 

•33 

•23 

•90 

•53 

•18 

17 

2-72 

•66 

•79 

•58 

2-22 

32 

5-10 

6-02 

•86 

•37 

•30 

47 

•49 

•39 

•90 

•09 

•17 

18 

2-88 

•82 

•80 

•59 

2-02 

33 

•26 

•18 

•86 

•52 

•29 

48 

•64 

•55 

•90 

•84 

•16 

19 

3-04 

•97 

•81      -63 

1-87 

34 

•42 

•34 

•87 

•67 

•28 

49 

•80 

•71 

•90 

8-00 

•16 

20 

3-20 

3-13 

•81      -73 

•76J 

35 

•58 

•49    -87 

•82 

•26 

50 

•96 

•86 

•90 

•96 

•16 

21 

3-35 

•29    -82 

•83 

•68 

36 

•74 

•65!  '87 

•97 

•25 

51 

8-12 

8-02 

•91 

•31 

•16 

22 

3-51 

•44 

•82 

•95 

•61 

37 

•90 

•81    '88 

6-13 

•24 

52 

•28 

•18 

•91 

•47 

•15 

23 

3-67 

•60 

•83 

4-07 

•56 

38 

6-05 

•97 

•88 

•29 

•23 

53 

•44 

•34 

•91 

•63 

•15 

24 

3-83 

•76 

•83 

•21 

•51 

39 

•21 

6-13 

•88 

•44 

•23 

54 

•60 

•50 

•91 

•79 

•14 

25 

3-99 

•91 

•84 

•34 

•47 

40 

•37 

•28 

•88 

•60 

•22 

55 

•76 

•66 

•91 

•95 

•14 

26 

4-15 

4-07 

•84 

•48 

•44 

41 

•53 

•44    -89 

•75 

•211 

56 

•92 

•81 

•91    9-10 

•14 

MACHINE  DESIGN  AND   MECHANICAL   CONSTRUCTIONS. 


281 


A 

"    ® 

SMALLEST   PINION,               .cj               «a 

SMALLEST   PINION, 

1  a 

£               SMALLEST   PINION, 

+3 

1 

O  -g 

TWELVE   TEETH. 

3 

-S 

-1 

TWELVE   TEETU. 

1 

<^1                   TWELVE    TEETH. 

•s 

x 

41 

Eadii  of  the  Kadii  of  the 
faces  of         flanks  of 

o 

11 

Radii  of  the 
faces  of 

Eadii  of  the 
flanks  of 

* 

0- 

§  -a     Eadii  of  the 
32        faces  of 

Eadii  of  the 
flanks  of 

& 

IB. 

teeth. 

teeth. 

1 

«'a 

teeth. 

teeth. 

to 

(g  ^         teeth. 

teeth. 

67 

9-08 

8-97 

•91 

9-26      '13 

120  19-10 

18-99 

19-10 

183   i>9  13  29-00    '97 

29-27 

68 

•23 

9-13 

•91 

•421     '13 

121 

•26  19-15; 

•42      '06 

184       -28      -16 

•43 

1-03 

69 

•37 

•29 

•92 

•67      '13 

122 

•42 

•30    -95 

•67 

185  !      -44      -32 

•59 

60 

•55 

•45 

•92 

•73      -13   123 

•58 

•46 

•73 

•06 

J186  I     '60      -48    -97 

•74 

1-03 

61 

•71 

•61 

•92 

•89 

•12 

124 

•74 

•62 

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187  i     -76  j     -64 

•90 

62 

•87      -77 

•92 

10-05 

•12 

125 

•90 

•78 

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20-05 

•06 

188       -921     -80 

30-06 

63 

10-03      -92 

•92 

•20 

•12 

126 

20-05 

•94 

•21 

189  30-08      -96    '98 

•22 

1-03 

64 

•19 

10-08 

•92 

•36 

•12 

127 

•21 

20-10 

•37 

•05 

190       -24  30-12 

•38 

65 

•35 

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•12  128 

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•40!     '28    -98      '54 

1-02 

66 

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192       -55i     -43         1     -70 

67 

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193       -71      -59 

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21-00 

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194       -87      '75    -98 

31-02 

1-02 

69 

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11-15 

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21-00 

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195  31-03      '91; 

•18 

70 

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133 

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73 

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200       -83      -71    -98 

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75 

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12-10      -10;  138 

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201       -99      -87 

32-13 

76  J12-10 

11-99 

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22-12 

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202  32-15  32-02| 

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j  203       -30!     -18 

•45 

78 

•42 

•30 

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•58 

•09   141 

•44 

•33 

•59 

204       -46      -34 

•61 

79 

•58      -47 

•74 

•09  j  142 

•60 

•48    '96 

•75      '05  205       -62!     '50 

•77 

80 

•73      -63 

•93 

•90 

•09,1143 

•76 

•64 

•91 

i  206 

•78      -66} 

•92 

81 

•89 

•79 

13-06 

•09!  144 

•92 

•80 

23-07      '05  207 

•94      -82          33-08 

S2 

13-05 

•94 

•93 

•22 

•09  145  123-08 

•96    '96 

•23 

208 

33-10      -98 

•24 

83 

•21 

13-10 

•38 

•09  i  146 

•24 

23-12 

•38|   1-04  209 

•2633-14 

•40 

84 

•37 

•26 

•94 

•53 

•08!  147 

•40 

•28J 

•54 

1210 

•42 

•30 

•56 

S5 

•53 

•42 

•94 

•69      -08   148 

•56 

•44! 

•70      -04  211 

•58 

•46 

•72 

86 

•69 

•58 

•85      -08 

1149 

•72 

•60    '96 

•86 

212 

•74 

•61 

•88 

87 

•85 

•74 

•94 

14-01 

•08 

J150 

•87 

•76 

24-02      '04  213 

•90 

•77 

34-04 

88 

14-01 

•90 

•94 

•17 

•08 

151 

24-03 

•92 

•18 

214 

34-06 

•93 

•20 

«9 

•17 

14-06 

•33 

•08 

152 

•19 

24-07    -96 

•34] 

215 

•21  34-09 

•36 

90 

•33 

•22 

•94 

•49 

•08  153 

•35 

•23 

•501     -04  216 

•37      -25 

•51 

91 

•49 

•38 

•94 

•65 

•08  154 

•51 

•39 

•65 

217       -53 

•41 

•67 

92 

•64 

•53 

•81 

•08   155 

•67 

•55    '96 

•81 

•04  218       -69 

•57 

•83 

93 

•80 

•69 

•94 

•97 

•08  156 

•83 

•71 

•98| 

219 

•85 

•73 

•99 

94 

•96 

•85 

•94 

15-12 

•07  157 

•99 

•87 

25-13 

•04  220 

35-01 

•89 

35-15 

96 

15-14 

15-01 

•30 

•07'  158 

25-15 

25-03    '97 

•29 

221 

•17  35-05 

•31 

96 

•28 

•17 

•94 

•44 

•07 

159 

•31 

•19 

•45      -04  222 

•331     -20 

'47 

J7 

•44 

•33 

•60 

M)7    160 

•47 

•35 

•61 

223 

•49      -36 

•63 

98 

•60 

•49 

•94 

•76 

•07   161 

•62 

•51    '97 

•77 

!  224 

•65      -52 

•79 

99 

•76 

•65 

•92      -07  1  162 

•78 

•66 

•93 

•04  225 

•80 

•68 

•95 

100 

•92 

•81 

•95 

16-08      -07   163 

•94 

•82 

26-09 

226 

•96 

•84 

36-10 

101 

16-08 

•97 

•24      -07  164 

26-10 

•98    -97 

•25 

•04  227 

36  12  36-00 

•26 

102 

•24 

16-13 

•40 

165 

•26 

26-14 

•42 

228        -28      '16 

•42 

103 

•39 

•28 

•95 

•56 

•07  166 

•42 

•30 

•56 

•04  229       '44      '32 

•58 

104 

•55 

•44 

•72 

*  167 

•58 

•46 

•72 

Ii230 

•59      -48 

•74 

105 

•71 

•60 

•87 

•07 

168 

•74 

•62    -97      '88 

•03H231 

•75      -64 

•90 

106 

•87 

•76    '9517-03 

169 

•90 

•78          27-04 

1232 

•91      -79 

37-06 

107 

17-03 

•92              -19      '06 

170 

27-06 

•94              -22 

•03II233  37-08 

•95 

•22 

108 

•19 

17'08              -35 

171 

•2227-10    -97      '36 

'234       -2437'H 

•58 

109 

•35 

•24    -95      -51      -06 

172 

•38!     -25 

•52      -03   235       -40 

•27 

•54 

110 

•51 

•40 

•67 

173 

•53      -41 

•68 

236  I     -56 

•43 

•69 

111 

•67      '56 

•83      -06  174 

•69 

•57 

•97      '84 

i  237 

•72 

•59 

•85 

112 

•83      -71 

•96 

•99 

175 

•85 

•73 

1-00 

•03  238 

•87 

•75 

38-01 

113 

•99 

•87 

18-15 

•06 

176 

28-01 

•89 

28-16 

239 

38-03 

•91 

•17 

114 

18-15 

18-03 

•95 

•30 

177 

•17 

28-05 

•97 

•31 

•03  240 

•19 

38-07 

•33 

115 

•30      -19 

•46 

•06 

178 

•33 

•21 

•47 

241 

•35 

•23 

•69 

116 

•46 

•35 

•26 

179 

•48 

•37 

•63 

!  242 

•51 

•38 

•85 

117 

•62 

•51 

•96      '62 

•06 

180 

•64 

•53 

•97 

•79      '03  243 

•67!     -54 

39-01 

118 

•78 

•67 

•78 

181 

•80 

•69 

•95 

244 

•83      -70 

•17 

119       -94 

•83 

•95 

•94 

1-06 

182       -97      '84l 

29-11     1-03  '245 

•99      '86 

•33 

282 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


No.  of  teeth. 

Radius  of 
pitch-circle. 

SMALLEST   PINION, 
TWELVE   TEETH. 

No.  of  teeth. 

Radius  of 
pitch-circle. 

SMALLEST  PINION, 
TWELVE   TEETH. 

No.  of  teeth. 

Radius  of 
pitch-circle. 

SMALLEST   PINION, 
TWELVE   TEETH. 

Radii  of  the 
faces  of 
teeth. 

Radii  of  the 
flanks  of 
teeth. 

Radii  of  the 
faces  of 
teeth. 

Radii  of  the 
flanks  of 
teeth. 

Radii  of  the    Radii  of  the 
faces  of          flanks  of 
teeth.              teeth. 

246 

39-15 

3902 

39-28 

265 

42-1742-04 

42-31 

284 

45-19 

45-06          45-33 

247 

•31 

•18 

44 

266 

•33      -20 

•46 

285 

•35 

.22 

•49 

248 

•47 

•34 

•60 

267 

•49      '36 

•62 

286 

•51 

•38 

•64 

249 

•64 

•50 

•76 

268 

•64      '52 

•78 

287 

•67 

•54 

•80 

250 

•78 

•86 

•92 

269 

•80      '68 

•94 

288 

•83 

•70 

•96 

251 

•94 

•82 

40-08 

270 

•97      '84 

43-10 

289 

•99 

•86 

46-12 

252 

40-10 

•97 

•24 

271 

43-13 

1-00 

•26 

290 

46-15 

46-02 

•28 

253 

•26 

40-13 

•40 

272 

•28 

43-15 

•42 

291 

•30 

•17 

•44 

254 

•42 

•30 

•56 

273 

•44 

•31 

•58 

292 

•46 

•33 

•60 

255 

•59 

•45 

•72 

274 

•60 

•47 

•74 

'293 

•62 

•49 

•76 

256 

•74 

•61 

•87 

275 

•76 

•63 

•90 

294 

•78 

•65 

•82 

257 

•90 

•77 

41-03 

276 

•92 

•79 

44-05 

295 

•94 

•81 

•98 

258 

41-06 

•93 

•20j 

277 

44*08 

•96 

•21 

296 

47-10 

•97 

i47'13 

259 

•22 

41-09 

•36 

278 

•24 

44-11 

•37 

297 

•25 

47-13 

•29 

260 

•38 

•25 

•51 

279 

•40 

•27 

•53 

298 

•42 

•29 

•45 

261 

•53 

•41 

•67 

280 

•55 

•43 

•69 

299 

•58 

•45 

•61 

262 

•69 

•56 

•83 

281 

•71 

•59 

•85 

300 

•74 

•61 

•77 

263 

•85 

•72 

•99 

282 

•87 

•74 

45-01 

Rak 

•129 

1-000-129 

1-00 

264 

42-01 

•89 

42-15 

283 

45-03 

•90 

•17 

Rule. — Seek  in  the  first  column  of  the  table  for  the  number  of  teeth  it  is 
proposed  that  the  wheel  shall  contain.  In  a  line  with  such  number  of  teeth 
take  from  columns  2,  3,  4,  5,  and  6  the  numbers  that  are  in  them  ;  and  in 
every  case  multiply  such  numbers  by  the  pitch.  The  products  will  be  the 
number  of  inches  and  parts  of  inches  to  which  the  compasses  must  be  opened 
to  describe  the  circles  and  parts  of  circles  that  are  required. 

Example. — Suppose  that  a  wheel  is  to  be  made  to  contain  thirty  teeth,  and 
that  the  pitch  of  the  teeth  is  to  be  2|  inches,  proceed  as  follows  :  Seek  in  col- 
umn 1  for  30,  the  number  of  proposed  teeth,  and  take  from  column  2  the 
numbers  4*783,  which  multiply  by  2£  inches,  the  product  will  be  11"*957.  Open 
the  compasses,  therefore,  to  this  radius  and  describe  a  circle,  which  will  be  the 
"  pitch-circle."  On  an  arc  of  this  circle  lay  off  2*5  X  '48  =  1*2"  for  the'thick- 
ness  of  a  tooth,  and  2*5x  5'2  —  1*3"  for  the  space.  Having  determined  the 
number  of  teeth  and  pitch,  next,  in  column  3,  and  in  the  same  line  with  30 
teeth,  will  be  found  the  numbers  4*704,  which  multiply  by  2%  inches — the 
product  will  be  11*75.  With  the  compasses  opened  to  this  distance,  and  from 
the  same  center  as  the  last,  describe  another  circle,  which  will  be  the  paths  of 
centers  for  the  curves  of  the  faces  of  the  teeth.  From  column  4  similarly  take 
the  numbers  0*865  and  multiply  by  &J  inches.  The  product  is  2*15,  to  which 
distance  the  compasses  must  be  opened  to  describe  the  faces  of  the  teeth. 

Again,  in  column  5,  multiply  5*07  X  2*5  =  12"-  675,  and  from  the  center, 
with  this  radius,  describe  another  circle  for  the  paths  of  centers  of  flanks  of  the 
teeth,  from  column  6,  1*34  X  2*5  =  3*35,  the  radius  of  the  flanks  of  the  teeth. 

For  the  height  of  a  tooth  a  common  proportion  is  T3¥  of  pitch  outside  of 
pitch-circle,  and  -fa  of  pitch  within,  which  leaves  -^  pitch  for  clearance  at  the 
bottom,  where  usually  small  arcs  are  described  to  connect  the  teeth  with  the 
wheel. 

Having  described  a  few  teeth  of  any  gear  to  its  full  size,  the  rest  may  belaid 
off  from  a  templet,  or  cutters  made  by  which  the  teeth  may  be  accurately 


MACHINE   DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


283 


formed.     In  the  illustration  (Fig.  579)  the  teeth  and  spaces  are  proportioned 
to  a  common  form,  but  there  is  considerable  variation  in  proportion,  as — 

Thickness  of  teeth,  from  '45  to  -48  pitch. 

Space  between  teeth,  from  -55  to  '52  pitch. 

Height  of  teeth  outside  of  pitch-circle,  from  '2  to  *3  pitch. 

Depth  of  teeth  inside  of  pitch-circle,  from  *3  to  *4  pitch. 


rH-ir- 


~~ _ 

(  i  """"-—•*. 


-  I 

sift 

'i 
I  ii 


FIG.  579. 


It  is  not  uncommon  to  make  one  of  the  set  of  gears  with  wooden  teeth, 
mortices  being  cast  in  the  periphery  of  the  wheel  for  the  insertion  of  these 
teeth — hence  called  mortise  wheels — the  elasticity  of  the  wood  diminishes  the 
effect  of  shocks,  and  they 
run  with  less  noise. 

The  usual  proportions 
and  construction  of  mor- 
tise wheels  are  shown  in 
Fig.  580,  a  section  across 
and  with  the  rim  of  the 
wheel.  The  figures  rep- 
resent the  proportions  to 
pitch  as  unity ;  b  is  from 
2  to  3  p.  The  teeth  are 
held  in  position  by  wood- 
en dovetailed  keys. 

Fig.  581  is  a  section 
across  the  rim  of  mortised 
bevel-gear ;  the  figures  are 
as  before  in  ratios  to  p. 
In  this  illustration  the 
teeth  are  held  in  by  pins, 
not  unusual  also  in  spur- 
mortise-gears. 

It  is  unusual  in  drawings  to  complete  gears  with  teeth  according  to  the  ex- 
amples given  ;  it  is  sufficient  for  the  purposes  of  pattern-making  that  the  pitch- 
circle,  pitch  and  form  of  one  tooth  be  given.  For  lines  of  shafting,  spur-gears 


FIG.  580. 


FIG.  581. 


284  MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 

may  be  represented,  like  plain  pulleys,  of  the  diameters  of  the  pitch-circle, 
with  the  pitch  and  number  of  teeth  written  in  :  bevel-gears,  as  in  Fig.  582. 
But,  as  in  finished  drawings  all  the  detail  is  necessary,  we  proceed  to  give  the 

simplest  forms  of  describing  spur-  and  bevel-gears 
with  sufficient  accuracy  for  all  practical  purposes. 

Projections  of  a  Spur- Wheel — To  draw  side  ele- 
vation (Fig.  583),  an  edge  view  (Fig.  584),  and  a 
*  vertical  section  (Fig.  585)  of  a  spur-wheel  with  34 

teeth  and  a  pitch  of  two  inches  : 

Determine  the  radius  of  the  pitch-circle  from  the 

U table,  page  278  ;  draw  the  central  line  A  C  B  and 

>      the  perpendicular  D  E  ;  on  C  as  a  center,  with  a 

U  radius  17*19,  describe  the  pitch-circle,  and  divide  it 

FIG.  582.  into  54  equal  parts.     To  effect  this  division,  with- 

out fraying  by  repeated  trials  that  part  of  the  paper 

on  which  the  teeth  are  to  be  represented,  describe  from  the  same  center  c, 
with  any  convenient  radius,  a  circle  abed  ;  with  the  same  radius  divide  its 
circumference  into  six  equal  parts,  and  subdivide  each  sixth  into  nine  equal 
parts,  and  draw  radii  to  the  center  c ;  these  radii  will  cut  the  pitch-circle  at 
the  required  number  of  points.  Divide  the  pitch  (2  inches)  into  10  equal 
parts ;  mark  off  beyond  the  pitch-circle  a  distance  equal  to  3  of  these  parts, 
and  within  it  a  distance  equal  to  4  parts,  and  from  the  center  0  describe  cir- 
cles passing  through  these  points  ;  these  circles  are  projections  of  the  cylinders 
bounding  the  points  of  the  teeth  and  the  roots  of  the  spaces  respectively. 

In  forming  the  outlines  of  the  teeth,  the  radii,  which,  by  their  intersections 
with  the  pitch-circle,  divide  it  into  the  required  number  of  parts,  may  be  taken 
as  the  center  lines  of  each  tooth.  The  thickness  of  the  tooth,  measured  on  the 
pitch-circle,  is  '46  pitch,  and  the  width  of  .the  space  is  equal  to  '54  p.  These 
distances  being  set  off,  take  in  the  compasses  the  length  of  the  pitch,  and  from 
the  center  g  describe  a  circular  arc  h  i ;  and  from  the  center  /,  with  the  same 
radius,  describe  another  arc  lik  touching  the  former  ;  these  arcs,  being  termi- 
nated at  the  circles  bounding  the  points  of  the  teeth  and  the  bottoms  of  the 
spaces  respectively,  form  the  curve  of  one  side  of  a  tooth.  The  other  side  is 
formed  in  a  similar  manner,  by  drawing  from  the  center  I  the  arc  m  n,  and 
from  the  center^  the  arc  mo,  and  so  on  for  all  the  rest  of  the  teeth. 

The  teeth  having  been  thus  completed,  we  proceed  to  the  delineation  of  the 
rim,  arms,  and  eye  of  the  wheel.  The  thickness  of  the  rim  is  usually  made 
equal  to  that  of  the  teeth,  say  ^  of  the  pitch,  which  distance  is  accordingly  set 
off  on  a  radius  within  the  circle  of  the  bottoms  of  the  spaces,  and  a  circle  is 
described  from  the  center  C  through  the  point  q  thus  obtained.  Within  the 
rim,  a  strengthening  feather  q  r,  in  depth  about  £  of  the  thickness  of  the  rim, 
is  generally  formed,  as  shown  in  the  plate.  The  eye,  or  central  aperture  for 
the  reception  of  the  shaft,  is  then  drawn  to  the  specified  diameter,  as  also  the 
circle  representing  the  thickness  of  metal  round  the  eye,  which  is  usually  made 
equal  to  the  pitch  of  the  wheel. 

To  draw  the  arms,  from  the  center  C,  with  the  radius  C  u  equal  to  the 
pitch,  describe  a  circle  ;  draw  all  the  radii,  as  C  L,  which  are  to  form  the  cen- 


MACHINE  DESIGN  AND   MECHANICAL  CONSTRUCTIONS.  285 

\ 


Fio.  534. 


286  MACHINE  DESIGN   AND   MECHANICAL  CONSTRUCTIONS. 

ter  lines  of  the  arms,  and  set  off  the  distance  L  v,  equal  to  £  pitch,  on  each  side 
of  these  radii  at  the  inner  circumference  of  the  rim  ;  and  through  all  the  points 
thus  obtained  draw  tangents  to  the  circle  passing  through  u.  The  contiguous 
arms  are  rounded  oft'  into  each  other  by  arcs  of  circles,  whose  centers  are  ob- 
tained by  the  following  construction  :  Taking,  for  example,  the  arc  M  P  Q,  it 
is  obvious  that  its  center  is  situated  in  the  straight  line  0  E  which  divides 
equally  the  interval  between  two  contiguous  arms.  Having  fixed  the  point  P 
(which  should  be  at  the  same  distance  from  t  as  the  breadth  of  the  feather  at 
the  back  of  the  rim)  draw  through  it  a  perpendicular  K  P  to  the  line  C  E  ;  the 
question  now  becomes  simply  a  geometrical  problem,  to  draw  a  circle  touching 
the  three  straight  lines  M  N,  P  R,  and  S  Q.  Divide  the  angle  P  R  M  into  two 
equal  parts  by  the  straight  line  R  0,  which  cuts  C  E  in  the  point  0,  the  center 
of  the  circle  required  ;  its  radius  is  the  line  0  M  perpendicular  to  M  N.  If 
now  a  circle  be  drawn  from  the  center  C  with  the  radius  C  0,  its  intersection 
with  the  radii  bisecting  all  the  intervals  between  the  arms  will  give  the  remain- 
ing centers,  such  as  0',  of  the  arcs  required  ;  and  the  circle  passing  similarly 
through  M  marks  all  the  points  of  contact  M  Q  M',  etc.  To  draw  the  small 
arcs  terminating  the  extremities  of  the  arms,  set  off  upon  the  line  C  E,  within 
the  point  r,  the  required  radius  of  the  arcs,  and  from  the  center  C  with  a 
radius  C  w  describe  a  circle  ;  the  distance  r  w  being  then  transferred  to  the 
extremities  of  the  arms  at  the  points  where  they  are  cut  by  the  circle,  as  at  Sx, 
will  give  the  centers  of  the  arcs  required.  Draw  the  central  web  of  the  arm  by 
lines  parallel  to  their  radii,  making  the  thickness  about  f  inch  for  wheel  of 
about  this  size. 

Having  thus  completed  the  elevation,  the  construction  of  the  edge  view  and 
vertical  section  becomes  comparatively  simple.  Draw  the  perpendiculars  F  G 
and  H  I  (Figs.  584  and  585)  as  central  lines  in  the  representations  ;  set  off  on 
each  side  of  these  lines  half  the  breadth  of  the  teeth,  and  draw  parallels  ;  pro- 
ject the  teeth  of  Fig.  583  upon  Fig.  584,  by  drawing  through  all  the  visible 
angular  points  straight  lines  parallel  to  A  B,  and  terminated  at  either  extremity 
by  the  verticals  representing  the  outlines  of  the  breadth  of  the  wheel ;  project 
in  like  manner  the  circles  of  the  hub  ;  lay  off  half  length  on  each  side  of  F  G, 
and  draw  parallels  to  it.  The  section  (Fig.  585)  is  supposed  to  be  made  on  the 
line  D  E  of  the  elevation  ;  project,  as  in  Fig.  584,  those  portions  which  will  be 
visible  in  this  section,  and  shade  those  parts  which  are  in  section.  The  arms 
are  made  tapering  in  width,  and  somewhat  less  than  the  face  of  the  wheel. 

Since  the  two  projections  (Figs.  583  and  585)  are  not  sufficient  to  exhibit 
fully  the  true  form,  a  cross-section  of  one  of  them  is  given  at  Fig.  586  ;  this 
section  is  supposed  to  be  made  by  a  plane  passing  through  X  X'  and  Y  Y'. 
The  points  y,  z,  in  Fig.  583,  and  corresponding  lines  in  Fig.  585,  represent  the 
edges  of  key-seat. 

Oblique  Projection  of  a  Spur-Wheel. — In  drawing  a  spur-wheel  or  other  ob- 
ject in  an  oblique  position  with  respect  to  the  vertical  plane  of  projection,  it  is 
necessary,  in  the  first  place,  to  lay  down  the  elevation  and  plan  as  if  it  were 
parallel  to  that  plane,  as  represented  in  Figs.  587  and  589.  Then  transfer  the 
plan  to  Fig.  590,  giving  it  the  same  inclination  with  the  ground  line  which  the 
wheel  ought  to  have  in  relation  to  the  vertical  plane  ;  and  assuming  that  the 


MACHINE   DESIGN  AND   MECHANICAL   CONSTRUCTIONS.  287 

fe 


288  MACHINE  DESIGN  AND   MECHANICAL   CONSTRUCTIONS. 

horizontal  line  A  B  represents  the  axis  of  the  wheel,  both  in  the  parallel  and 
oblique  positions,  the  center  of  its  front  face  in  the  latter  position  will  be  de- 
termined by  the  intersection  of  a  perpendicular  raised  from  the  point  C'  (Fig. 
590)  with  that  axis.  Now,  it  is  obvious  that  if  we  take  any  point,  as  a  in  Fig. 
587,  the  projection  of  that  point  on  Fig.  589  must  be  in  the  line  a  a,  parallel  to 
A  B ;  and  further,  this  point  being  projected  at  a'  (Fig.  590),  it  must  be  in 
the  perpendicular  a'  a  ;  therefore  the  intersection  of  these  two  lines  is  the  point 
required.  Thus  all  the  remaining  points  b,  c,  d,  etc.,  may  be  obtained  by  the 
intersections  of  the  perpendiculars  raised  from  the  points  b ',  c',  d',  etc.  (Fig. 
590)  respectively,  with  the  horizontals  drawn  through  the  corresponding  points 
in  Fig.  587.  It  will  also  be  observed  that  since  the  points  e  and/,  in  the  fur- 
ther face  of  the  wheel,  have  their  projections  in  a  and  b  (Fig.  587),  their  oblique 
projections  will  be  situated  in  the  lines  a  a  and  b  b,  but  they  are  also  at  e  and 
/;  consequently,  the  lines  ea  and  fb  are  the  oblique  projections  of  the  edges 
a'  e'  and  b'f.  We  have  now  to  remark  that  all  the  circles  which,  in  the  rec- 
tangular elevation  (Fig.  587),  have  been  employed  in  the  construction  of  this 
wheel  are  projected  in  the  oblique  view  into  ellipses,  the  length  and  position 
of  whose  axes  may  be  determined  without  any  difficulty  ;  for  since  the  plane 
F'  G',  in  which  these  circles  are  situated,  is  vertical,  the  major  axes  of  all  the 
ellipses  in  question  will  obviously  be  perpendicular  to  the  line  A  B,  and  equal 
to  the  diameters  of  the  circles  of  which  they  are  respectively  the  projections  ; 
and  the  minor  axes,  representing  the  horizontal  diameters,  will  all  coincide 
with  the  line  A  B.  Thus,  to  obtain  the  ellipse  into  which  the  pitch-circle 
is  projected,  it  is  only  necessary  to  set  off  upon  the  vertical  D  E  (Fig.  589), 
above  and  below  the  point  C,  the  radius  of  the  pitch-circle,  whose  horizontal 
diameter  ij  being  at  i'f  (Fig.  590)  is  projected  to  ij  (Fig.  589)  ;  and  thus 
having  obtained  the  major  and  minor  axes,  the  ellipse  in  question  may  easily 
be  constructed.  The  intersection  of  the  horizontal  lines  g  g,  hh,  etc.,  with 
this  circle  gives  the  thickness  of  the  teeth  at  the  pitch-line  ;  and,  by  projecting 
in  the  same  manner  the  circles  bounding  the  extremities  and  roots  of  the  teeth, 
these  points  in  each  individual  tooth  may  be  determined  by  a  similar  process. 
If  strict  accuracy  is  required,  a  greater  number  of  points  is  necessary  for  the 
construction  of  the  curvature  of  the  teeth,  and  two  additional  circles  m  n  and 
op  maybe  drawn  on  Fig.  587,  and  projected  to  Fig.  589,  and  the  points  of  their 
intersection  with  the  curves  of  the  teeth  projected  to  Fig.  589,  where  the  cor- 
responding points  are  indicated  by  the  same  letters. 

Projections,  of  a  Bevel -Wheel. — Fig.  591  is  a  face  view,  Fig.  592  an  edge 
view,  and  Fig.  593  a  vertical  transverse  section.  For  the  determination  of  the 
division  of  the  angle  of  inclination  of  the  axes  of  a  pair  of  bevel- wheels,  see 
Fig.  575)  ;  for  their  size  and  proportion,  the  rules  given  for  spur-wheels  ;  thus, 
consider  the  base  of  the  cone  A  B  (Figs.  592  and  593)  as  the  diameter  of  the 
pitch-circle  of  a  spur-wheel,  and  proportion  the  pitch,  form,  and  breadth  of 
teeth,  according  to  the  stress  to  which  they  are  to  be  subjected. 

Having  determined  and  laid  down,  according  to  the  required  conditions, 
the  axis  0  S  of  the  primitive  cone,  the  diameter  A  B  of  its  base,  the  angle 
A  S  0  which  the  side  of  the  cone  makes  with  the  axis,  and  the  straight  lines 
A  o,  D  0',  perpendicular  to  A  S,  and  representing  the  sides  of  two  cones,  be- 


MACHINE  DESIGN   AND  MECHANICAL   CONSTRUCTIONS.  289 


290  MACHINE  DESIGN  AND   MECHANICAL  CONSTRUCTIONS. 

tween  which  the  breadth  of  the  wheel  (or  length  of  the  teeth)  is  comprised, 
the  first  operation  is  to  divide  the  primitive  circle,  described  with  the  radius 
A  C,  into  a  number  of  equal  parts  corresponding  to  the  number  of  teeth  or 
pitch  of  the  wheel.  Then  upon  the  section  (Fig.  593)  draw  with  the  radius 
o  A  or  o  B,  supposed  to  move  parallel  to  itself,  outside  the  figure,  a  small  por- 
tion of  a  circle,  upon  which  construct  the  outlines  of  a  tooth  M,  and  of  the  rim 
of  the  wheel,  with  the  same  proportions  and  after  the  same  manner  as  we  have 
explained  in  reference  to  spur-wheels  ;  set  off  from  A  and  B  the  points  a,  d, 
and/,  denoting  respectively  the  distances  from  the  pitch-line  to  the  points  and 
roots  of  the  teeth,  and  to  the  inside  of  the  rim,  and  join  these  points  to  the 
vertex  S  of  the  primitive  cone,  terminating  the  lines  of  junction  at  the  lines 
D  o',  E  o' ;  the  figure  abed  will  represent  the  lateral  form  of  a  tooth,  and  the 
figure  cdfe  a  section  of  the  rim  of  the  wheel,  by  the  aid  of  which  the  face 
view  (Fig.  591)  may  easily  be  constructed. 

The  points  a,  b,  c,  d,  and  e,  having  been  projected  upon  the  vertical  diam- 
eter A'  B',  describe  from  the  center  C'  a  series  of  circles  passing  through  the 
points  thus  obtained,  and  draw  any  radius,  as  C'  L,  passing  through  the  center 
of  a  tooth.  On  either  side  of  the  point  L  set  off  the  distances  L  &,  L  I,  making 
up  the  thickness  of  the  tooth  M  at  the  point,  and  indicate,  in  like  manner, 
upon  the  circles  passing  through  the  points  B'  and  d',  its  thickness  at  the  pitch- 
line  and  root;  then  draw  radii  through  the  points  i,  I,  Tc,  g,  m,  etc.,  termi- 
nating them  respectively  at  the  circles  forming  the  projections  of  the  corre- 
sponding parts  at  the  inner  extremity  of  the  teeth  ;  these  radial  lines  will  repre- 
sent the  rectilinear  edges  of  all  the  teeth.  The  curvilinear  outlines  may  be 
delineated  by  arcs  of  circles,  tangents  to  the  radii  g  C'  and  i  C',  and  passing 
through  the  points  obtained  by  the  intersections  of  the  radii  and  the  various 
concentric  circles.  The  radii  of  these  circular  arcs  may  in  general,  as  in  the 
case  of  spur-wheels,  be  taken  equal  to  the  pitch,  and  their  centers  upon  the 
interior  and  exterior  pitch-circles  ;  thus  the  points  g  and  i,  n  and  o,  for  exam- 
ple, are  the  centers  for  the  arcs  passing  through  the  corresponding  points  in 
the  next  adjacent  teeth,  and  vice  versa. 

The  drawing  of  the  teeth  in  the  edge  view  (Fig.  592),  and  of  such  portions 
of  them  as  are  visible  in  the  section  (Fig.  593),  is  sufficiently  explained  by  in- 
spection of  the  lines  of  projection  introduced  into  the  plate  for  this  purpose. 
In  the  construction  of  these  views,  observe  that  every  point  in  the  principal 
figure  from  which  they  are  derived  is  situated  upon  the  projection  of  the  circle 
drawn  from  the  center  0',  and  passing  through  that  point.  Thus  the  points 
g  and  i,  for  example,  situated  upon  the  exterior  pitch-circle,  will  be  determined 
in  Fig.  592  by  the  intersection  of  their  lines  of  projection  with  the  base  A  B  of 
the  primitive  cone  ;  and  the  points  Tc  and  I  will  be  upon  the  straight  line 
passing  through  a  a  (Fig.  593),  and  so  on.  Farther,  as  the  lateral  edges  of 
all  the  teeth  in  Fig.  591  are  radii  of  circles  drawn  from  the  center  C',  so  in 
Fig.  592  they  are  represented  by  lines  drawn  through  the  various  points  found 
as  above  for  the  outer  extremities  of  the  teeth,  and  converging  toward  the 
common  apex  S  ;  while  the  center  lines  of  the  exterior  and  interior  extremities 
themselves  all  tend  to  the  points  o  and  o'  respectively. 

Skew-Bevels. — When  the  axes  of  wheels  are  inclined  to  each  other,  and  yet 


MACHINE   DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


291 


do  not  meet  in  direction,  and  it  is  proposed  to  connect  them  by  a  single  pair  of 
bevels,  the  teeth  must  be  inclined  to  the  base  of  the  frusta  to  allow  them  to 
come  into  contact.  Set  off  a  e  (Fig.  594)  equal  to  the  shortest  distance  between 
the  axes  (called  the  eccentricity],  and  divide  it  in  c,  so  that  a  c  is  to  e  c  as  the 
mean  radius  of  the  frustum  to  the  mean  radius  of  that  with  which  it  is  to  work  ; 
draw  c  m  d  perpendicular  to  a  e.  The  line  c  m  d  gives  the  direction  of  the  teeth  ; 
and,  if  from  the  center  «,  with  radius  a  c,  a  circle  be  described,  the  direction  of 
any  tooth  of  the  wheel  will  be  a  tangent  to  it,  as  at  c.  Draw  the  line  d  e  per- 
pendicular to  c  m  d,  and  with  a  radius  d  e  equal  to  c  e  describe  a  circle ;  the 
direction  of  the  teeth  of 
the  second  wheel  will  ~"~~ 
be  tangents  to  this  last, 
as  at  d. 

System  composed  of  a 
Pinion  driving  a  Rack 
(Fig.  595).— The  pitch- 
line   M  N  of  the  rack 
and  the  pitch-circle  A 
B  D  of  the  pinion  being 
laid  down  touching  one 
another,  divide  the  lat- 
ter into  twice  the  num- 
ber of  equal  parts  that 
it  is  to  have  of  teeth,  and  set  off  the 
common  distance  of  these  parts  upon 
the  line  M  N,  as  many  times  as  may 
be  required  ;   this  marks  the  thick- 
ness of  the  teeth  and  width  of  the 
spaces  in  the  rack.     Perpendiculars 
drawn  through  all   these   points   to 
the  solid  part  of  the  rack  will  rep- 
resent the  flanks  of  the  teeth  upon 
which  those  of  the  pinion  are  to  be 
developed  in  succession.     The  curva- 
ture of  these  latter  should  be  an  in- 
volute A  c  of  the  circle  A  B  D.     The 
teeth  might  be  cut  off  at  the  point  of 
contact  d  upon  the  line  M  N,  for  at 

this  position  the  tooth  A  begins  its  action  upon  that  of  the  rack  E  ;  but  it  is 
better  to  allow  a  little  more  length ;  in  other  words,  to  describe  the  circle 
bounding  the  points  of  the  teeth  with  a  radius  somewhat  greater  than  C  d. 

With  regard  to  the  form  of  the  spaces  in  the  rack,  all  that  is  required  is  to 
set  off  from  M  1ST,  as  at  the  point  e,  a  distance  slightly  greater  than  the  differ- 
ence A  a  of  the  radius  of  the  pitch-circle,  and  that  of  the  circle  limiting  the 
points  of  the  teeth,  and  through  this  point  to  draw  a  straight  line  F  G  parallel 
to  M  N.  From  this  line  the  flanks  of  all  the  teeth  of  the  rack  spring,  and  their 
points  are  terminated  by  a  portion  of  a  cycloid  A  b,  which,  however,  may  in 


FIG.  594. 


292  MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


— 

E 

0 

VJ 

P 

c -- 


FIG.  595. 


most  instances  be  replaced  by  an  arc  of  a  circle.     The  depth  of  the  spaces  in 
the  pinion  obviously  depends  upon  the  height  of  this  curved  portion  of  the 


FIG.  596. 


MACHINE   DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


293 


teeth  ;  their  outline  is  formed  by  a  circle  drawn  from  the  center  C,  with  a 
radius  a  little  less  than  the  distance  from  this  point  to  the  straight  line  bound- 
ing the  upper  surface  of  the  teeth  of  the  rack. 

System  composed  of  a  Rack  driving  a  Pinion. — In  this  case  the  construc- 
tion is  in  all  respects  identical  with  that  of  the  preceding  example,  with  this 
exception,  that  the  form  proper  to  be  given  to  the  teeth  of  the  rack  is  a  cycloid 
generated  by  a  point  A  in  the  circumference  of  the  circle  AEG  rolling  on  the 
line  M  N.  The  curvature  of  the  teeth  is  an  involute  as  before. 

System  composed  of  an  Internal  Spur-  Wheel  driving  a  Pinion  (Fig.  596). — 
The  form  of  the  teeth  of  the  driving-wheel  is  in  this  instance  determined  by 
the  epicycloid  described  by  a  point  in  the  circle  A  E  0,  rolling  on  the  concave 
circumference  of  the  primitive  circle  M  A  N.  The  points  of  the  teeth  are  to 
be  cut  oif  by  a  circle  drawn  from  the  center  of  the  internal  wheel,  and  passing 


FIG.  597. 


through  the  point  E,  which  is  indicated,  as  before,  by  the  contact  of  the  curve 
with  the  flank  of  the  driven  tooth. 

The  wheel  being  supposed  to  be  invariably  the  driver,  the  curved  por- 
tion of  the  teeth  of  the  pinion  may  be  very  small.      This  curvature  is  a 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


part  of  an  epicycloid  generated  by  a  point  in  the  circle  M  A  N  rolling  upon 
BAD. 

System  composed  of  an  Internal  Wheel  driven  by  a  Pinion  (Fig.  597). — 
This  problem  involves  a  different  mode  of  treatment  from  that  employed  in 
the  preceding  cases.  The  epicycloidal  curve  A  a,  generated  by  a  point  in  the 
circle  having  the  diameter  A  0,  the  radius  of  the  circle  M  A  N,  and  which 
rolls  upon  the  circle  BAD,  can  not  be  developed  upon  the  flank  A b,  the  line 
described  by  the  same  point  in  the  same  circle  in  rolling  upon  the  concave  cir- 
cumference MAN;  and  for  this  obvious  reason,  that  that  curve  is  situated 
without  the  circle  BAD,  while  the  flank,  on  the  contrary,  is  within  it.  It 
becomes  necessary,  therefore,  in  order  that  the  pinion  may  drive  the  wheel 
uniformly  according  to  the  required  conditions,  to  form  the  teeth  so  that  they 
shall  act  always  upon  one  single  point  in  those  of  the  wheel.  This  may  be 
most  advantageously  effected  by  taking  for  the  curvature  of  the  teeth  of  the 
pinion  the  epicycloid  A  d,  described  by  the  point  A  in  the  circle  MAN  rolling- 
over  the  circle  BAD.  It  will  be  observed  that,  as  in  the  preceding  examples, 
the  tooth  E  of  the  pinion  begins  its  action  upon  the  tooth  F  of  the  wheel  at 
the  point  of  contact  of  their  respective  primitive  circles,  and  that  it  is  un- 
necessary that  it  should  be  continued  beyond  the  point  c,  because  the  succeed 
ing  tooth  H  will  then  have  been  brought  into  action  upon  G  ;  consequent!} 
the  teeth  of  the  wheel  might  be  bounded  by  a  circle  passing  through  the  point  c. 
It  is,  however,  one  of  the  practical  advantages  which  this  species  of  gearing  has 
over  wheels  working  externally  that  the  surfaces  of  contact  of  the  wheel  and 
pinion  admit  of  being  more  easily  increased  ;  and,  by  making  the  teeth  some- 
what longer  than  simple  necessity  demands,  the  strain  may  be  distributed  over 
two  or  more  teeth  at  the  same  time.  The  flanks  of  the  teeth  of  the  wheel  are 
formed  by  radii  drawn  to  the  centre  0,  and  their  points  are  rounded  off  to  en- 
able them  to  enter  freely  into  the  spaces  of  the  pinion. 


DRAWING   OF   SCKEWS. 


Projections  of  a  Triangular-threaded  Screw  and  Nut  (Fig.  598). — Having 
drawn  the  ground  line  A  B,  and  the  center  lines  C  C'  of  the  figures,  from  0  as 
a  center,  with  a  radius  equal  to  that  of  the  exterior  cylinders,  describe  the 
semicircle  a  3  6  ;  describe  in  like  manner  the  semicircle  bee  with  the  radius  of 
the  interior  cylinder.  Now  draw  the  perpendiculars  a  a"  and  6  6",  b  b"  and  e  e", 
which  will  represent  the  vertical  projections  of  the  exterior  and  interior  cylin- 
ders. Then  divide  the  semicircle  a  3  6  first  described  into  any  number  of  equal 
parts,  say  6,  and  through  each  part  draw  radii,  which  will  divide  the  interior 
semicircle  similarly.  On  the  line  a'  a"  set  off  the  length  of  the  pitch  as  many 
times  as  may  be  required  ;  and  through  the  points  of  division  draw  straight 
lines  parallel  to  the  ground  line  A  B.  Then  divide  each  distance  or  pitch  into 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


295 


twice  the  number  of  equal  parts  that  the  semicircles  have  been  divided  into, 
and,  following  instructions  already  laid  down  (page  102),  construct  the  helix 
a'  3'  6  both  in  the  screw  and  nut. 

Having  obtained  the  point  b' ',  by  the  intersection  of  the  horizontal  line  pass- 
ing through  the  middle  division  of  a'  a  with  the  perpendicular  b  b",  describe  the 
helix  b'  c'  e',  which  will  represent  the  bottom  of  the  groove.  The  apparent  out- 


G 
FIG.  599. 

lines  of  the  screw  and  its  nut  will  then  be  completed  by  drawing  the  lines  b'  af, 
a'  b'9  etc. ,  to  the  curves  of  the  helices  ;  these  are  not,  strictly  speaking,  straight 
lines,  but  their  deviation  from  the  straight  line  is,  in  most  instances,  so  small 
as  to  be  imperceptible,  and  it  is  therefore  unnecessary  to  complicate  the  drawing. 
When  a  long  series  of  threads  have  to  be  delineated,  they  should  be  drawn 
mechanically,  by  means  of  a  mold  or  templet  constructed  in  the  following 


296 


MACHINE  DESIGN  AND   MECHANICAL  CONSTRUCTIONS. 


manner  :  Take  a  small  slip  of  thin  wood  or  pasteboard,  and  draw  upon  it  the 
helix  a'  3'  6  to  the  same  scale  as  the  drawing,  and  pare  the  slip  carefully 
and  accurately  to  this  line.  By  applying  this  templet  upon  Fig.  598,  so 
that  the  points  a'  and  6  on  the  plate  shall  coincide  with  a'  and  6  on  the  draw- 
ing, the  curve  a'  3'  6  can  be  drawn  mechanically,  and  so  on  for  the  remain- 
ing curves  of  the  outer  helix.  The  same  templet  may  be  employed  to  draw 
the  corresponding  curves  in  the  screw-nut  by  simply  inverting  it ;  but  for  the 
interior  helix  a  separate  one  must  be  cut,  its  outlines  being  laid  off  in  the  same 


manner. 


Projections  of  a  Square-threaded  Screw  and  Nut  (Fig.  599). — The  depth 
of  the  thread  is  equal  to  its  thickness,  and  this  latter  to  the  depth  of  the  groove. 
The  construction  is  similar  to  the  preceding,  and  will  be  readily  understood 
from  the  drawing,  the  same  letters  and  figures  marking  relative  parts.  The 
parts  of  the  curve  concealed  from  view  are  shown  in  dotted  lines. 


TJ1  ------- 


FIG.  600. 

System  composed  of  a  Wheel  and  Tangent,  or  Endless  Screw.  — In  laying  out 
the  work,  the  pitch  of  the  teeth  is  to  be  determined  by  the  stress,  as  for  spur- 
wheels,  and  the  number  of  the  teeth  in  the  wheel  by  the  number  of  turns  of 
the  screw  for  each  revolution  of  the  wheel.  Suppose  these  determined,  and  0 
(Fig.  600)  to  be  the  center  of  the  wheel,  E  F  the  axis  of  the  screw,  C  A  the 
radius  of  the  pitch-circle  of  the  wheel,  and  G  A  that  of  the  pitch-cylinder  of 
the  screw  ;  the  line  M  N  drawn  through  A,  parallel  to  E  F,  will  be  the  gen- 
eratrix of  that  cylinder,  which  will  serve  the  purpose  of  determining  the  form 
of  the  teeth.  The  section  is  made  through  the  axis,  and  is  obviously  the  case 
of  a  rack  driving  a  pinion  ;  consequently  the  curve  of  the  teeth,  or  rather 
thread,  of  the  screw  should  be  simply  a  cycloid  generated  by  a  point  in  the  cir- 
cle AEG,  described  upon  A  C  as  a  diameter,  and  rolling  upon  the  straight  line 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


297 


M  N.  The  outlines  of  the  teeth  are  helical  surfaces  described  about  the  cylin- 
der forming  the  screw,  with  the  pitch  A  b  equal  to  the  distance,  measured  upon 
the  primitive  scale,  between  the  corresponding  points  of  two  contiguous  teeth. 
These  curves  are  expressed  by  dotted  lines.  The  teeth  of  the  wheel  are  set  at 
angle  to  the  plane  of  its  face,  and  with  surfaces  corresponding  to  the  inclination 
and  helical  form  of  the  thread  of  the  screw.  Usually  the  points  of  the  teeth 
and  bottoms  of  the  spaces  are  formed  of  a  concave  outline,  adapted  to  the  con- 
vexity of  the  screw,  in  order  to  present  as  much  bearing  surface  as  possible  to 
its  action.  In  this  kind  of  gearing  it  is  invariably  the  screw  that  imparts  the 


FIG.  601. 


FIG.  602. 


motion  ;  but  in  the  proportions  adopted  by  the  Yale  &  Towne  Manufacturing 
Co.  for  worm  gearing,  the  wheel  under  the  weight  will  revolve  the  screw  slowly. 
This  angle  of  the  teeth  is  found  to  be  the  best  adapted  for  economy  of  power. 
In  the  wheel  the  teeth  in  section  are  those  of  a  spur-wheel,  cut  with  a  chasing 
cutter,  and  in  the  screw  turned  in  a  lathe. 

Figs.  601  and  602  are  two  views,  worm  and  wheel,  with  such  lines  of  con- 
struction dotted  as  will  explain  the  manner  of  drawing. 

functional  Gearing. — When  motion  is  not  continuous  for  along  time,  either 
having  frequently  to  be  stopped  and  started  or  reversed,  frictional  gearing  is 


298 


MACHINE   DESIGN  AND   MECHANICAL  CONSTRUCTIONS. 


very  often  used.  The  starting  is  with  as  little  shock  as  with  belting,  and  un- 
der the  proper  conditions  of  pressure  it  is  fully  as  positive,  and  by  the  usual 
appliances  this  pressure  may  be  applied  gradually.  The  simplest  form  of  fric- 

tional  gearing  is  that  in  which  the  surfaces  in 
contact  correspond  to  that  of  the  pitch-circles 
(Fig.  603). 

Fig.  604  is  a  bevel  frictional  gear,  such  as  is 
used  in  Dow's  grain  stores,  Brooklyn,  N.  Y.  One 
half  is  shown  in  section.  The  surface  of  the 
upper  or  larger  gear  is  of  cast-iron,  that  of  the 
lower  of  paper,  in  washers  compressed  by  a  hy- 
draulic press  and  firmly  held  together  by  bolts. 
The  bevel  in  section  is  in  contact  with  the  large 

wheel-surface,  the  other  is  disengaged.  A  slight  motion  to  the  right  will  throw 
out  that  in  contact,  and  not  throw  in  the  other,  and  motion  ceases  in  the 
large  driven  wheel ;  a  still  further  motion  throws  in  the  left  pinion,  and  the 
motion  of  the  driven  wheel  is  reversed. 


FIG.  603. 


The  mode  in  which  this  is  done  is  shown  in  Fig.  605.  The  shipper  consists 
of  a  bell-crank,  controlled  by  a  screw.  The  screw  works  in  a  stand,  on  the  top 
of  which  is  a  hand  wheel ;  the  hand  wheel  can  be  moved  in  either  direction, 
and  any  desirable  pressure  can  be  brought  upon  the  frictional  surfaces  by  means 
of  the  screw.  It  is  not  unusual,  instead  of  two  pinions  to  have  one  pinion, 
with  a  little  clearance  on  each  side,  revolving  between  two  wheels,  a  slight 
lateral  motion,  in  either  direction,  bringing  it  in  contact  with  one  or  the  other 
of  the  wheels.  Some  provision,  by  a  loose  coupling  or  otherwise,  must  be  made 
to  admit  of  this  lateral  movement  in  the  pinion  shaft.  Straight  pulleys,  or 
what  would  correspond  to  spur-gears  without  teeth,  are  constructed,  as  in  the  ex- 
ample given,  and  are  thrown  in  or  out  of  gear  by  a  lateral  motion  of  the  pinion. 

In  proportioning  the  face  of  the  pulleys  it  has  been  found  safe  to  consider 
it  the  same  as  belts,  given  in  the  table  (page  273).  The  pressure  can  be 
applied  according  to  the  requirements  of  driving,  and  there  is  no  falling  off  in 
the  friction.  The  frictional  surfaces  are  not  always  paper  ;  wood,  leather,  and 
prepared  rubber  are  frequently  used. 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


299 


Wedge   Gearing,  or  Robertson   Grooved- Surf  ace 
Frictional  Gearing. — Fig.  606  is  the  cross-section  of 
the  rims  of  two  wheels  of  this  gearing.     The  angle 
recommended  by  Robertson  is  50°  (usually  not  over 
30°  in  our  practice),  and  the  pitch  to  vary  somewhat 
with  the  velocity  and  power  to  be  transmitted.    The 
adhesion,  under  a  pressure  equal  to  that  of  the  ten- 
sion of  a  belt,  is  proved  to  be  greater,  and  it  would 
be  safe  to  make  the 
horizontal  face  equal 
to  that   of    a    belt 
under  the  same  cir- 
cumstances of  trans- 
fer of  power. 


FIG.  605. 


The  use  of  ropes  as  belts  has  been  treated  of  (page  274),  but  they  are  often 
used,  as  in  Fig.  607,  for  a  reciprocating  power.     The  ropes  are  not  endless, 

but  consist  of  two  ropes,  the  ends  of  which  are 
attached  to  two  drums  parallel  with  each  other, 
each  having  several  turns  on  the  barrels  or 
drums,  but  in  opposite  directions,  so  that,  by 
the  motion  of  the  drums,  one  rope  will  un- 
wind from  one  drum  and  wind  up  on  the  oth- 
er, and  vice  versa,  the  length  of  the  recipro- 
catory  movement  being  measured  by  the  turns 
on  one  of  the  drums.  This  arrangement  is 
FIG.  606.  sometimes  applied  to  run  the  barrels  of  a  hoist ; 

the  barrels  being  attached  to  one  drum  and 

the  power  applied,  at  the  other,  and  in  this  form  the  application  may  be  at 
considerable  distance  apart. 


FIG.  607. 


A  similar  arrangement  with  chains,  instead  of  ropes,  was  much  used  for  the 
reciprocating  motion  of  the  bed  in  the  older  type  of  planers. 


300  MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 

The  following  table  is  from  "Appletons'  Cyclopaedia  of  Applied  Mechanics" 

TABLE  SHOWING  EOPES  ANT)  CHAINS  OF  EQUAL  STRENGTH. 


SIZES,   IN   INCHES,   FOE   EQUAL   8TBENGTH. 

AVERAGE   WEIGHT  PEE  FOOT. 

Working 
Strain. 

Crucible 
Steel  Eope. 

Charcoal 
Iron  Eope. 

Hemp  Rope. 

Iron  Chain. 

Steel  Hope. 

Iron  Eope. 

Hemp  Eope. 

Jron  Chain. 

Cir. 

Cir. 

Cir. 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Tons. 

.... 

I'OO 

2f 

•h 

0   14 

0'34 

0-50 

0-3 

.... 

1-18 

3 

i 

.... 

0-21 

0-46 

0-65 

0-4 

•00 

1-39 

H 

3E2 

0-17 

0-28 

0-67 

0-81 

0-5 

•26 

1-57 

*J 

A 

0-25 

0-33 

0-75 

0-96 

0-6 

•45 

1-77 

4i 

1 

0-30 

0  45 

0-83 

1-38 

0-8 

57 

1-97 

5 

A 

0-35 

0-57 

1-16 

1-76 

1-0 

•77 

2-19 

«i 

if 

0-45 

0'70 

1-20 

2-20 

1  3 

1-96 

2-36 

6f 

i 

0-59 

0-83 

1-60 

2-63 

1-5 

2-36 

2-75 

6f 

f 

0-85 

1-08 

2-00 

4*21 

2-3 

2'75 

3-14 

n 

tt 

1-10 

1-43 

2'65 

4-83 

3-1 

2-95 

3  •  53 

8f 

£ 

T28 

1-80 

3'35 

5-75 

3'8 

3-14 

3-93 

9f 

* 

1-45 

2'30 

4-00 

7'50 

4-8 

3'53 

4-32 

10| 

if 

1-83 

2-94 

4  92 

9-33 

5-9 

3-93 

4-71 

111 

ih 

2'33 

3-56 

5-83 

10-6 

7-0 

4'32            5-10 

12£ 

H 

2  98 

4-00 

6-20 

11-9 

8-2 

4-71            5-50 

14f 

H 

3-58 

4-80 

8-70 

14-5 

9-5 

4-81 

5-89 

15* 

if 

3'65 

5  60 

9-00 

17-6 

11-0 

5-10 

6'28 

15| 

H 

4  04 

6-30 

10-1 

20*0 

12-5 

5-89 

7-07 

17* 

if 

5-65 

7'95 

18  7 

22-3 

15-9 

6-35 

7-85 

m 

if 

6  50 

9'81 

16-4 

24-3 

19-6 

Endless  chains  are  often  used  for  the  transmission  of  power,  where  the 
stress  is  great  and  the  movement  slow.  When  the  chain  used  is  of  the  com- 
mon form,  the  wheels  must  be  fitted  with  depressions  or  caps  to  receive  the 
flat  links,  with  a  slot  for  the  vertical  links,  as  in  Fig.  617.  A  chain  com- 


1 

: 

~i         P 

J 

. 

1                1 

1             1 

I     i    1 

1 

1 

I 

i         r 

1 

n          r 

:      !    J 

1 

J           U 

1 

r  "T 

J 

FIG. 


posed  of  punched  links,  as  in  Fig.  608,  admits  of  a  tooth  between  the  links, 
and  the  wheels  on  which  these  run  have  therefore  teeth  adapted  to  the  chain, 
which  is  composed  of  links  of  uniform  length. 

But  the  chief  application  of  ropes  and  chains  is  for  the  purpose  of  hoisting 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


301 


•or  lowering  heavy  weights  or  loads,  by  the  means  of  pulleys  and  blocks,  or  bar- 
rels and  capstans. 

Rope  for  running-rigging  is  usually  made  of  hemp  or  manilla,  and  wire- 
rope  for  this  purpose  is  mostly  made  with  hemp  centers. 

A  simple  rule  for  the  working-strength  of 
these  ropes  is  to  multiply  the  square  of  the 
girth  or  circumference  of  the  rope  by  100  for 
hemp  or  manilla,  600  for  iron- wire  rope,  and 


FIG.  609. 


FIG.  610. 


1,000  for  steel-wire  rope,  and  the  result  will  be  the  working-strength  in  pounds. 
Fig.    609  is  the  front  and  side  view  of  a  common  wooden  block,  iron- 
strapped.    The  pulley  or  sheave  is  shown  in  Fig.  610  ;  the  section  shows  a  bush- 
ing at  the  center  for  the  pin  ;  the  sheaves  are  of  lignum  vitae. 


FIG.  611. 


FIG.  612. 


FIG.  613. 


FIG.  614. 


Figs.  611,  612,  613,  614  are  wrought-iron  tackle-blocks  of  the  Yale  &  Towne 
Manufacturing  Company's  pattern.  The  lower  block  of  every  set  is  always 
sent  with  a  becket  attached,  as  shown  in  Fig.  612. 


Diameter  of  sheave 

In. 
21 

In. 

31 

In. 
4 

In. 

4| 

In. 
5 

i,. 

6& 

In. 

Y 

In. 

8 

In. 
q 

In. 
10 

In. 
11 

Will  take  rope   diameter. 

1 

1 

£ 

1 

1 

U 

u 

H 

9, 

91 

9,1 

Will  take  chain   diameter 

3 

1 

5 

A 

JL 

A 

4 

4 

Grin-blocks  (Fig.  615). — These  blocks  are  made  with  wrought-  and  malle- 
able-iron frames  and  wrought  swivel-hook. 


302 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


In.    1    In. 

In. 

In. 

In. 

In. 

In. 

Diameter  of  wheel 

10      12 

14 

16 

18 

90 

22 

Will  take  rope,  diameter  

1        1 

11 

H 

14- 

H 

li 

Ton.    Ton. 

Ton. 

Ton. 

Ton. 

Ton. 

Ton. 

Will  carry  about 

1       U 

11 

0, 

2i 

2i 

21 

* 

FIG.  615. 


Winding-drums  or  barrels  must  have  their  diameters  pro- 
portioned to  the  diameters  of  the  rope  or  chain  to  be  used  (see 
table  of  sheaves  above),  and  their  length  to  the  length  of  rope 
or  chain  to  be  taken  in,  and  when  the  coils  or  turns  of  the  rope 
are  numerous  provision  must  often  be  made  for  keeping  the 
rope  or  chain  so  that  one  coil  may  not  ride  on  another.  This  is  done  by  spiral 
grooves  in  the  barrel,  or  shifting  the  barrel  or  the  rope-guide  automatically. 

Fig.  1504  shows  the  way  in  which  a  chain  cable  is  taken  in  with  but  few  coils 
on  the  barrel.  The  coils  are  sufficient  for  the  friction  of  taking  up  the  cable  ; 
the  tight  cable  is  wound  on  the  larger  part  of  the  barrel,  and  as  the  coils  are 
unwound  on  the  slack  side  the  tight  coil  slips  down  to  a  smaller  diameter ; 
the  weight  of  the  chain  on  the  slack  side,  as  it  drops  into  the  locker,  is  suffi- 
cient to  preserve  the  friction  ;  but  with  a  rope,  a  man  takes  in  the  rope  and 
exerts  at  the  same  time  a  little  strain.  The  application  of  a  barrel  of  this  form 
for  hoisting  is  very  common  ;  by  exerting  a  slight  stress  the  man  can  hoist  a 
weight  on  a  revolving  barrel,  and  by  slacking  he  can  lower  without  changing 
the  direction  of  motion  or  speed  of  the  barrel. 

Chain-wheels  with  pockets,  which  have  been  spoken  of  in  their  application 
to  the  transmission  of  power,  are  also  especially  applicable  to  the  purpose  of 
hoisting,  requiring  a  width  only  slightly  greater  than  that  of  the  chain,  and 
a  diameter  sufficient  to  give  the  proper  engagement  with  it. 


FIG.  617. 


FIG.  616. 


Flat  punched  links  are  of  uniform  length,  and  can  be  purchased  of  any  de- 
sirable sizes,  and  put  together  in  multiples  ;  common  chain  has  not  that  uni- 
formity in  length  to  adapt  it  nicely  to  the  pockets  of  the  wheel.  The  Yale 


MACHINE  DESIGN  AND   MECHANICAL  CONSTRUCTIONS. 


303 


&  Towne  Manufacturing  Company  have  made  a  spiral  chain,  of  common  form 
but  of  uniform  length,  especially  adapted  to  hoists,  and  Figs.  616,  617,  and 
618  illustrate  the  construction  of  their  chain-wheel.  A  is  a  pocketed  chain- 
wheel,  made  of  soft  cast-iron,  mounted  on  a  frame  B.  0  is  the  chain-guide 
enveloping  the  lower  half  of  the  chain-wheel.  The  inner  curved  surface  of  the 


FIG.  619. 


FIG.  620. 


FIG.  621. 


FIG.  622. 


FIG.  623. 


FIG.  624. 


chain-guide  is  grooved,  and  is  of  such  a  shape  as  to  leave  a  space  between  it  and 
the  periphery  of  the  chain-wheel  merely  sufficient  to  admit  the  chain  ;  it  must 
then  enter  properly  and  continue  engaged  with  the  chain- wheel.  E  is  a  chain- 
guide  roller,  that  delivers  the  slack  chain  into  the  box  or  locker.  D  is  the 
chain-stripper,  bolted  also  to  the  plate  B,  with  a  tongue  or  rib  projecting  into 
the  center  groove  of  the  wheel  which  disengages  the  chain. 

The  usual  forms  of  chain-cables  are  represented  by  the  open  circular  link 
(Fig.  619),  the  open  oval  (Fig.  620),  oval  with  pointed  stud  (Fig.  621),  oval 
with  broad-headed  stud  (Fig.  622),  an  obtuse  angled  stud-link  (Fig.  623),  and 
the  parallel-sided  stud- link  (Fig.  624).  The  usual  proportions  of  chain-links  are 
6  diameters  of  the  iron  in  length 
by  3£  in  width.  The  end  links, 
which  terminate  each  15  fathoms 
of  chain,  are  6*5  in  length  to  4*1 
in  breadth,  and  the  iron  about 
1-2  the  diameter  of  the  rest  of 
the  chain. 

Hooks.—Fig8.    625    and   626 
(from    Redtenbacher)    represent 


two  wrought-iron  hooks,  in  which 
the  material  is  distributed  accord- 
ing to  the  strain  to  which  the 
parts  may  be  subjected.  The 
following  are  the  proportions  on 
which  Fig.  625  is  constructed : 
Assuming  the  neck  of  the  hook 
as  the  modulus  or  1,  the  diam- 
eter of  journals  of  the  traverse 
are  1*1 ;  width  of  traverse  at  center,  2  ;  distance  from  the  center  of  the  hook 
to  the  center  of  the  traverse,  7*5;  interior  circle  of  the  hook,  3*4;  greatest 


FIG.  625. 


FIG.  626. 


304: 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


FIG.  627. 


thickness  of  the  hook,  2  '8.  Assuming  (Fig. 
626)  the  diameter  of  the  wire  of  the  chain 
as  1  :  interior  circle  of  hook  is  3 '2,  and 
greatest  thickness  of  hook  3  *5. 

Fig.  627  represents  a  hook  as  made  by  the 
Yale  &  Towne  Manufacturing  Company. 
This  hook  is  fitted  in  a  cross-head  ;  the  diam- 
eter at  A  is  that  of  iron  from  which  the  hook 
is  forged,  and  the  section  shown  hatched  at 
the  center  of  the  hook  is  equal  to  that  of  the 
round  iron. 

It  has  been  shown  that  hooks,  of  the  pro- 
portions but  with  a  much  greater  load  than 
given  in  the  following  table,  yield  by  the 
gradual  opening  of  the  jaw,  giving  ample 
notice  before  rupture. 


Capacity  of  hook 

Ton. 

1 

Ton. 

i 

Ton. 
1 

Ton. 
1 

Ton. 

Ton. 

2 

Ton. 

3 

Ton. 

4 

Ton. 

5 

Ton. 
6 

Ton. 

8 

Ton. 
10 

Dimensions  of  A  

In. 

4 

In. 
if 

In. 

4 

In. 
1-iV 

In. 
11 

In. 

15- 

In. 

14 

In. 

2 

In. 
9,1 

In. 

<>4 

In. 

25- 

In. 
31 

Dimensions  of  D  

11 

1* 

H 

14 

9, 

9,1 

9,4 

31 

3£ 

41 

51 

61 

All  parts  of  the  hook  are  expressed  in  parts  of  A,  and  can  readily  be  de- 
termined from  the  scale  above. 

Figs.  628  and  629  are  side  and  front  elevations  of  an  ordinary  straight  lever 
on  a  shaft ;  both  are  shown  broken,  either  because  the 
length  is  indefinite,  or  because  it  is  inconvenient  to  put  on 
the  paper.  The  handle  should  be  from  5  to  6  inches  long, 
and  1^  diameter.  The  bar  beneath  the  handle  to  be  square, 
and  of  uniform  width  on  one  side  of  the  lever  and  a  taper 
on  the  other,  as  shown,  of  about  y  in  4  feet  on  each  side. 
The  sides  of  the  square  at  the  handle  to  be  i  |/  length  in 
inches,  or  say  ¥  for  30"  lever,  $"  for  4  feet,  and  1"  for  5 
feet.  The  neck  of  the  shaft  to  be,  as  proportioned  in  the 
drawing,  about  T^-  of  the  greatest  width  of  the  lever,  and 
the  diameter  of  hub  1^-.  The  stress  exerted  by  a  man  may 
be  from  75  to  100  pounds,  and  the  size  of  the  shaft  will 
depend  on  the  torsion al  stress  between  the  hub  of  the  lever 
and  the  point  of  resistance. 

Fig.  630  is  a  hand-lever  forming  one  arm  of  a  bell-crank 
— a  bolt  passing  through  a  slot  in  the  frame  and  the  arm  of 
the  lever,  and  the  two  are  clamped  together  by  a  thumb- 
FIG.  628.    FIG.  629.      nut,  n,  by  which  the  lever  can  be  held  in  any  position. 
The   same   purpose   is   often   effected    by  notches    in   the 
frame,  into  which  the  arm  of  the  lever  is  caught,  or  by  spring  latches,  as  in 
Fig.  631, 


MACHINE   DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


305 


Figs.  632  and  633  are  side  view  and  plan  of  a  foot-lever.  The  foot-plate  is 
8"  X  5"  X  f ",  and  as  the  lever  is  subject  to  double  the  stress  of  the  hand-lever 
above,  the  dimension  should  be  somewhat  increased.  The  side  of  square  next 


FIG.  630. 


FIG.  631. 


FIG.  632.         FIG.  633. 


the  foot-plate  should  be,  say  for  a  lever  of  30",  V  ;  of  4  feet,  1^  ;  of  5  feet,  1£; 
the  form  and  taper  as  in  the  hand-lever. 

Figs.  634  and  635  are  views  of  a  hand-crank.    The  diameter  of  the  handles, 
for  convenience  in  grasping,  should  not  be  less  than  1-J"  ;  if  for  the  force  of  two 
men,  l-j-",  and  from  the  diameter  of  the  handle  the  rest  may  be  proportioned  as 
in  the  figure.    The  length 
of  handle  for  a  single  man 
should  be  from  10"  to  12" ; 
for  two  men,  from  20  to 
24  :  the  crank  from  15"  to 


FIG.  634. 


FIG.  635. 


18",  and  the  height  of  shaft 
above  the  foot  support  for 
the  men  from  2' 10"  to  3' 2". 

Engine  -  Cranks.  — Fig. 
636  is  a  graphic  represen- 
tation made  from  a  table 
from  Bourne's  "  Handbook 
of  the  Steam-Engine,"  for 
determining    "the    diame- 
ters of  wrought  crank-shaft  journals" — i.  e.,  of  the  large  eye  of  the  crank. 
The  ordinates  are  diameters  in  inches  of  the  steam  cylinder,  the  inclined  lines 
the  stroke  in  feet,  and  the  abscissas  the  diameters  of  the  eye  in  inches. 

Use  of  the  Table. — To  find  the  diameter  of  large  eye  of  crank  of  a  steam- 
engine  40"  cylinder  and  4-foot  stroke.  Find  on  what  line  of  abscissa  is  the  in- 
tersection of  the  ordinate  40"  with  the  diagonal  4'  of  stroke,  which  will  be 
about  S-J",  the  diameter  of  crank-eye. 

20 


306 


MACHINE   DESIGN  AND   MECHANICAL   CONSTRUCTIONS. 


The  table  is  calculated  on  a  steam  pressure  in  the  cylinder  of  25  pounds  ; 
not  the  average  pressure,  but  the  maximum.  This  pressure  is  much  less  than 
present  practice,  but  the  table  can  be  readily  adapted  to  any  pressure.  For 

Diameter  of  Eye. 


Ao'  ~" 

jj  —  .  

~  ~  «•  =  *  *  *----;;=-"  -  "\^\\\"  "  ~tj^ 



i.  r~    ~  ~      ---           -             .,» 

'""""       -\~T  "  '"       _-r-r  "       j_«--' 

1                       •""        ~  "       "        ~^~ie:"-l 

_--="'       ----""i---  I"       "    ~ 

,"r-,ijf£  "_--*'"    ;SE3"^*«E 

_--  £;;  =  ;;  "                                             -    -  - 

u  EE»;;:^=  •;:!;=!:::::£=::: 

if  \]^"^^^^^^?^\\^\\\  \\\\\\ 

::!•=!;;"•=;:  

l=:::: 

20" 

jo*          w*          .w          £0 

;0'          (9/r       tt5^™ 

Diameter  of  Steam-Cylinder  in  Inches. 
FIG.  636. 

most  stationary  engines  the  pressure  is  from  75  to  100  pounds  ;  for  75,  the  area 
on  diagram  must  be  three  times  what  it  is  for  25  pounds.  Thus,  for  a  steam- 
cylinder  of  30"  diameter  and  under  75  pounds  pressure,  multiply  the  area  of 

75 

30"D.  X  25  =  706-9  X  3=2120'7  =  area  of  52"  diameter,  which  use  for  determin- 
ing the  diameter  of 
eye  instead  of  30". 
It  will  agree  very 
nearly  with  com- 
mon practice  for 
stationary  engines 
to  multiply  the  di- 
ameter of  cylinder 
in  diagram  by  2,  for  the  diameter 
to  be  used,  and  for  locomotives,  by 

For  the  small  eye  of  the  crank, 
under  the  same  conditions  of  pres- 
sure, Bourne  gives  the  rule  :  Multi- 
ply diameter  of  cylinder  by  -142. 
This  is  too  small  for  the  present  prac- 
tice, which  is  from  -17  to  '25  or  -J 
to  \  the  diameter  of  the  cylinder. 
The  crank-pins  are  made  of  steel  or  FIG.  63Y.  FIG.  638. 

iron  case-hardened.     Eyes  are  bored 

by  hydraulic  or  screw  press  to  a  very  tight  fit,  and  forced  on  to  the  shaft  or 
pin,  or  heated  and  shrunk  on. 

Figs.  637  and  638  are  two  views  of  a  wrought-iron  crank,  and  Figs.  639 
and  640  of  a  cast-iron  crank,*  both  proportioned  in  their  parts  to  the  diameter 

*  "  Elements  of  Machine  Design,"  Unwin. 


MACHINE  DESIGN   AND  MECHANICAL   CONSTRUCTIONS. 


307 


.5  G.U 


of  the  large  eye 
as  unity,  but,  as 
shown  by  the  di- 
agram and  rule 
following,  these 
figures  can  only 
apply  to  a  single 
throw  of  crank, 

as  the  diameters  of  the  two  eyes 
vary  as  their  distances  apart. 

Taking  the  diameter  of  the  large 
eye  of  the  crank,  Eedtenbacher 
gives  in  the  table  the  relative  sizes 
of  central  and  end  eyes  of  cranks, 
depending  on  the  proportion  be- 
tween the  length  of  crank  and  the 
diameter  of  central  eye.  The  first 
column  exhibits  the  number  of 
times  the  diameter  of  eye  is  con- 
tained in  the  length  of  crank  ;  the 
second  and  third  columns  give  the  suitable  diameters  of  crank-pins. 

Figs.  641  and  642  represent  a  side  and  front  elevation  of  a  crank,  such  as 


FIG.  639. 


FIG.  640. 


is  used  on  engines  of  American  river  boats. 


The  main  body  of  the  crank  is  of 
cast-iron,  with  two  horns  a  a 
projecting  from  the  central  hub, 
and  the  whole  is  bound  with  a 
strap  of  wrought-iron. 

DIAMETEE  OF  EYE,  BEING  UNITS. 


-tr 


FIG.  641. 


For  wrought- 
iron  shafts. 

Cast-iron 
shafts 

2 

0  85 

0-62 

3 

0-69 

0'51 

4 

0-60 

0-44 

5 

0-54 

0-39 

6 

0-49 

0-36 

7 

0-45 

0-33 

8 

0-42 

0-31 

9 

0-40 

0-29 

10 

0-38 

0-28 

11 

0-36 

0  26 

12 

0-34 

0-25 

10 

0-33 

0-24 

FIG.  642. 

The  diameters  of  crank-pins 

as  above  given  are  on  the  basis  of  a  length  of  from  1  to  1-J  of  the  diameter ; 
if  the  length  be  increased  beyond  this  the  diameter  should  be  increased  in  the 
ratio  of  1  to  the  square  root  of  the  diameter. 


308 


MACHINE   DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


Disk-cranks  are  circular  disks  of  cast-iron,  with  crank-pins  of  iron  or  steel, 
and  as  much  strength  of  metal  around  the  pin  as  in  the  crank.  They  are  bet- 
ter than  the  crank,  in  that  there  is  no  unbalanced  crank  and  pin,  and  part  of 


the  weight  of  the  connection  can  be  balanced  by  a  proper  disposition  of  metal 
within  the  area  of  the  disk. 

Fig.  643  is  a  plan  of  a  double  crank-axle,  although  by  the  projection  the 


FIG.  644. 


FIG.  645. 


FIG.  646. 


lower  axle  A  appears  as  a  straight  shaft.  The  dimensions  given  are  from  an 
axle  in  use.  In  construction  the  cranks  are  rectangular  in  section,  of  which 
the  width  is  •£$  the  depth,  and  the  depth  1*5  the  diameter  of  crank-journal 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


309 


Cranks  are  usually  forged  solid,  and  the  slot  for  the  crank  cut  out ;  that  shown 
in  the  figure  was  cast  in  steel  for  a  double  compound  engine,  7  X  15  X  15,  and 
although  there  is  often  great  condensation  in  the  cylinders,  it  has  worked 
satisfactorily  for  many  years. 

Eccentrics. — An  eccentric  is  a  modified  crank ;  the  crank-pin  is  enlarged  so 
as  to  include  the  crank-shaft ;  motion  is  conveyed  through  the  crank  to  the 
pin,  and  not  through  the  pin  to  the  shaft. 

Fig.  644  represents  a  front  view,  Fig.  645  the  side  view,  and  Fig.  646  a  sec- 
tion, of  a  form  of  eccentric  usually  adopted  in  steam-engines  for  giving  motion 
to  the  valves  regulating  the  action  of  the  steam  upon  the  piston.  A  ring  or 
hoop,  eccentric  strap,  is  accurately  fitted  within  projecting  ledges  on  the  outer 
circumference  of  the  eccentric,  so  that  the  latter  may  revolve  freely  within  it ; 
this  ring  is  connected  by  an  inflexible  rod  with  a  system  of  levers,  by  which  the 
valve  is  moved.  It  is  evident,  that  as  the  shaft  to  which  the  eccentric  is  fixed 
revolves,  an  alternating  rectilinear  motion  will  be  impressed  upon  the  rod,  its 
amount  being  determined  by  the  eccentricity,  or  distance  between  the  center  of 
the  shaft  and  that  of  the  exterior  circle.  The  throw  of  the  eccentric  is  twice 
the  eccentricity  C  E  ;  or  it  may  be  expressed  as  the  diameter  of  the  circle  de- 
scribed by  the  point  E.  The  nature  of  the  alternating  motion  generated  by 
the  circular  eccentric  is  identical  with  that  of  the  crank. 


FIG.  647. 

Fig.  647  is  a  common  form  of  eccentric  strap  and  rod  adapted  to  the  draw- 
ing of  the  eccentric  given  ;  it  is  usually  fitted  with  a  composition  bush,  and  a 
pan  must  be  provided  beneath  to  catch  any  oil  that  may  drip  from  the  eccen- 
tric. This  last  may  be  avoided  by  the  use  of  an  eccentric  strap,  Figs.  648,  649, 
650,  in  which  it  will  be  seen  that  the  strap  forms  a  cup-section  (Fig.  650) 
which  secures  a  projecting  ring  on  the  eccentric,  and  retains  the  oil.  These 
figures  represent  the  eccentric  strap  of  a  locomotive,  and  are  made  entirely  of 
cast-iron  ;  the  bolts  are  very  long,  and  the  strap  exceedingly  rigid. 

In  practice,  the  term  eccentric  is  generally  confined  to  the  circular  eccen- 
tric ;  all  others,  with  exception  of  that  last  described,  being  called  cams  or 
wipers. 

Projections  of  Eccentrics. — The  term  eccentric  is  often  applied  in  general 
to  all  such  curves  as  are  composed  of  points  situated  at  unequal  distances  from 
a  central  point  or  axis. 


310 


MACHINE  DESIGN  AND  MECHANICAL  CONSTEUCTIONS. 


FIG.  650. 


FIG.  648. 


FIG.  649. 

Fig.  651. — To  draw  the  eccentrical  symmetrical  curve  called  the  heart,  which 
is  such  as,  when  revolving  with  a  uniform  motion  on  its  axis,  to  communicate 
to  a  movable  point  A,  a  uniform  rectilinear  motion  of  ascent  and  descent. 

Let  C  be  the  axis  or  center  of 
rotation  upon  which  the  eccentric  is 
fixed,  and  which  is  supposed  to  re- 
volve uniformly ;   and  let  A  A'  be 
the  distance  which  the  point  A  is 
s       required  to  traverse  during  a  half 
\       revolution  of  the  eccentric.     From 
\      the  center  C,  with  radii  respectively 

ft 


equal  to  C  A  and  C  A',  describe  two 
circles  ;  divide  the  greatest  into  any 
number  of  equal  parts  (say  16),  and 
draw  through  these  points  of  di- 
vision the  radii  01,  C  2,  03,  etc. 
Then  divide  the  line  A  A'  into  the 
same  number  of  equal  parts  as  are 
contained  in  the  semicircle  (that  is, 
into  8  in  the  example  now  before 
us),  and  through  all  the  points  1', 

2',  3',  etc.,  draw  circles  concentric  with  the  former  ;  the  points  of  their  inter- 
section B,  D,  E,  etc.,  with  the  respective  radii  C  1,  C  2,  C  3,  etc.,  are  points 
in  the  curve  required,  its  vertex  being  at  the  point  8. 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


311 


It  will  now.  be  obvious  that  when  the  axis,  in  its  angular  motion,  shall  have 
passed  through  one  division  ;  in  other  words,  when  the  radius  0  1  coincides 
with  C  A',  the  point  A,  being  urged  upward  by  the  curvature  of  the  revolving 
body  on  which  it  rests,  will  have  taken  the  position  indicated  by  1' ;  and  fur- 
ther, when  the  succeeding  radius  C  2  shall  have  assumed  the  same  position,  the 
point  A  will  have  been  raised  to  2',  and  so  on  till  it  arrives  at  A',  after  a  half 
revolution  of  the  eccentric.  The  remaining  half,  A  G  F  8,  of  the  eccentric, 
being  exactly  symmetrical  with  the  other,  will  enable  the  point  A  to  descend 
in  precisely  the  same  manner  as  it  is  elevated.  It  is  thus  manifest  that  this 
curve  is  fitted  to  impress  a  uniform  motion  upon  the  point  A  itself,  but  in 
practice  a  small  friction  roller  is  usually  interposed  between  the  surface  of  the 
eccentric  and  the  piece  which  is  to  be  actuated  by  it.  Accordingly,  the  point 
A  is  to  be  taken  as  the  center  of  this  roller,  and  the  curve  whose  construction 
we  have  just  explained  is  replaced  by  another,  similar  to  and  equidistant  from 
it,  which  is  drawn  tangentially  to  arcs  of  circles  described  from  the  various 
points  in  the  primary  curve  with  the  radius  of  the  roller.  This  second  curve 
is  manifestly  endowed  with  the  same  properties  as  the  other  ;  for,  supposing 
the  point  e,  for  example,  to  coincide  with  A,  if  we  cause  the  axis  to  revolve 
through  a  distance  equal  to  one  of  the  divisions  the  point/,  which  is  the  inter- 
section of  the  curve  with  the  circle  whose  radius  is  C  1',  will  then  obviously 
have  assumed  the  position  V ;  at  the  next  portion  of  the  revolution,  the  point  g 
(which  is  such  that  the  angle/ C  g  is  equal  to  e  C/)  will  have  arrived  at  2',  and 
so  on.  Thus  it  is  plain  that  the  point  a  will  be  elevated  and  depressed  uni- 
formly by  means  of  the  second  curve,  in  the  same  manner  as  that  denoted  by 
A  is  actuated  by  the  first. 

It  is  obvious  that  the  movable  point  a  must,  in  actual  working,  be  held 
in  contact  with  the  surface  of  the  ^ 

eccentric  ;  this  is  generally  accom- 
plished by  the  action  of  a  weight 
or  of  a  spring ;  but  in  forms  simi- 
lar to  Fig.  651,  in  which  all  the 
diameters,  as  A  A  8,  B  F,  D  G,  etc., 
are  equal,  two  frictions  connected 
and  placed  diametrically  opposite 
each  other  may  be  used,  which  will 
be  thus  alternately  and  similarly 
impelled  ;  in  many  cases  an  eccen- 
tric groove  is  cut,  and  the  friction 
roll  or  point  a  is  made  to  slide  in 
this  groove. 

Fig.  652. — To  draw  a  double  and 
symmetrical  eccentric  curve,  such  as 
to  cause  the  point  A  to  move  in  a 
straight  line,  and  with  an  unequal 

motion  ;  the  velocity  of  ascent  being  accelerated  in  a  given  ratio  from  the  start- 
ing-point to  the  vertex  of  the  curve,  and  the  velocity  of  descent  being  retarded 
in  the  same  ratio. 


FIG.  652. 


312 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


Upon  A  A'  as  a  diameter  describe  a  semicircle,  and  divide  it  into  any  num- 
ber of  equal  parts  ;  draw  from  each  point  of  division  1,  2,  3,  etc. ,  perpendicu- 
lars upon  C  A'  ;  and  through  the  points  of  intersection  1',  2',  3',  etc.,  draw 
circles  having  for  their  common  center  the  point  C,  which  is  to  be  joined,  as 
before,  to  all  the  points  of  division  on  the  circle  (A'  48).  The  points  of  inter- 
section of  the  concentric  circles  with  the  radii  01,  02,  03,  etc.,  are  points  in 
the  curve  required. 

Fig.  653. — To  construct  a  double  and  symmetrical  eccentric,  which  shall 
•produce  a  uniform  rectilinear  motion,  with  periods  of  rest  at  the  points  nearest 
to,  and  farthest  from,  the  axis  of  rotation. 

The  lines  in  the  figure  above  referred  to  indicate  sufficiently  plainly,  with- 
out the  aid  of  further  description,  the  construction  of  the  curve  in  question, 
which  is  simply  a  modification  of  the  eccentric  represented  at  Fig.  651.  In 
the  present  example,  the  eccentric  is  adapted  to  allow  the  movable  point  A  to 
remain  in  a  state  of  rest  during  the  first  quarter  of  a  revolution  B  D  ;  then, 


FIG.  653. 


FIG.  654.     FIG.  655. 


during  the  second  quarter,  to  cause  it  to  traverse,  with  a  uniform  motion,  a 
given  straight  line  A  A',  by  means  of  the  curve  D  G-  ;  again,  during  the  next 
quarter  E  F  G,  to  render  it  stationary  at  the  elevation  of  the  point  A' ;  and 
finally,  to  allow  it  to  subside  along  the  curve  B  E,  with  the  same  uniform  mo-' 
tion  as  it  was  elevated,  to  its  original  position,  after  having  performed  the  entire 
revolution. 

Fig.  654  represents  an  edge  view  of  this  eccentric,  and  Fig.  655  a  vertical 
section  of  it. 

If  but  one  side  of  this  were  constructed,  and  the  motion  only  equal  to  that 
of  the  arc  and  reciprocating,  it  would  raise  and  lower  every  point  resting  on  it, 
and  would  be  called  a  wiper.  The  wiped  surface  is  generally  flat,  an  arm 
extending  out  from  the  rod  to  be  raised,  and  a  curve  D  Gr  may  be  formed 
adapted  to  any  height  of  lift,  and  action  during  the  lift. 

Connections. — Figs.  656  and  657  are  sections  of  cottered  joints  of  wrought- 
iron  bars,  the  first  made  with  a  socket  and  the  end  of  one  of  the  bars  ;  the 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


313 


latter  by  a  sleeve  connecting  the  two  bars.  The  bars  in  the  socket  and  sleeve 
are  upset,  to  give  more  section  than  the  bars  themselves,  so  that  the  slots  cut 
for  the  cotters  c  c  will  not  reduce  the  strength  below  that  of  the  bars.  The 
cotters  must  have  sufficient  shearing  strength  and  bearing  surface,  and  at  the 


same  time  diminish  as  little  as  possible  the  section  of  the  parts  connected. 
The  proportions  given  in  the  figures  are  drawn  to  a  scale  of  the  diameter  of 
the  enlarged  part  as  the  unit,  and  the  proportions  given  in  figures  are  such  as 
obtain  in  practice  for  wrought-iron.  If  the  cotter  be  of  steel,  its  breadth  may 
be  f  of  that  given,  preserving  the  other  dimensions  the  same ;  the  thickness  is 
*25  of  the  unit. 

The  knuckle-joint  (Fig.  658)  is  given  in  dimensions  of  the  bar  as  a  unit, 
and  adapted  to  usual  work.     If  there  is  much  motion  at  the  joint,  the  wearing- 
surface  should  be  larger,  by  increasing 
the  width  of  the  eyes  and  the  length 
of  the  pin.     The  pin  in  the  drawing  is 
through  the  collar  ;  usually  the  pin  is 
extended,  and  the  pin  passes  through 
the  bolt  outside  the  collar. 

Connecting-rods,  in  their  applica- 
tion to  steam-engines,  are  the  rods 
connecting  the  piston  through  the 
cross-head  to  the  crank.  When  two 
cranks  are  connected  it  is  called  a 
coupling-rod. 

Figs.  659,  660,  and  661  are  side 
plan  and  end  views  of  a  connecting-  FIG.  658. 

rod,  as  made  by  the  South wark  Foun- 
dry, of  Philadelphia,  and  used  on  their  fast-running  Porter- Allen  engines. 

The  cross-head  end  is  a  strap-end,  while  that  of  the  crank  is  a  box-end,  and 
the  latter  is  made  of  larger  diameter  than  the  former  on  account  of  the 
application  of  the  stress  to  the  crank-pin,  and  the  wear,  this  pin  is  made  larger 
than  the  pin  of  the  cross-head.  The  length  of  the  page  does  not  admit  of  the 
representation  of  the  full  length  of  the  connecting-rod  on  the  scale ;  it  is 


314:  MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


MACHINE  DESIGN   AND   MECHANICAL  CONSTRUCTIONS. 


315 


therefore  shown  broken,  with  the  dimensions  figured  in.     The  sections  of  the 
two  ends  are  drawn  in  on  the  rods  ;  the  circular  section  A  is  the  same  as  that 


FIG.  662. 


of  the  piston-rod,  and  both  are  represented  in  the  conventional  hatching  of 
cast-iron.    This  is  of  wrought-iron.    The  gib  g  and  key  or  cotter  v  at  the  strap- 


FIG.  664. 


end  are  of  steel,  and  the  key  is  fastened  when  in  position  by  a  set-screw  through 
the  head.     At  the  box-end,  a  wedge  and  screw  forces  the  box  into  position. 


316 


MACHINE   DESIGN   AND  MECHANICAL  CONSTRUCTIONS. 


It  will  be  observed  on  the  plan  that  this  rod  is  drawn  as  though  it  were 
flat  on  top ;  but  as  the  tops  are  curved,  it  is  more  accurately  represented  in 
Fig.  662. 


FIG.  665. 


Fig.  663  is  a  strap-end  of  a  connecting-rod,  from  the  Corliss  Steam-Engine 
Company.  The  peculiarity  is  the  adjusting-screws  connected  with  the  boxes. 

Fig.  664  is  the  strap-end  of  a  locomotive  connecting-rod  in  which  the  wear 
of  the  boxes  is  taken  up  by  a  cotter  at  the  end  ot  the  strap. 


FIG.  667. 


In  Fig.  665  the  key  is  between  the  bolts  ;  the  weakness  from  bolt-holes  or 
cotter-slot  is  compensated  by  the  width  of  the  strap. 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS.  317 

Fig.  666  is  a  cast-iron  eccentric  strap  ;  the  bolts  are  very  long  and  the  con- 
nection very  rigid.  The  box  is  fitted  with  metalline,  which  is  put  in  small 
disks  ;  oiling  is  thereby  avoided. 

The  bolts  for  the  large  end  are  bored  up  for  the  greater  part  of  their  length, 
to  reduce  their  sectional  area  to  that  of  the  screwed  portion  and  thus  secure 
equal  elasticity  ;  with  these  long  bolts  no  check-nuts  are  necessary. 

In  many  marine  engines  the  boxes  of  both  crank  and  cross-head  pins  are 
made  similar  to  this,  with  the  bolts  strong  and  heavy,  and  connecting  the  two 
boxes  without  any  other  rod. 


FIG.  669. 


FIG.  670. 

Fig.  669  is  the  box-end  of  a  locomotive  ;  the  section  (Fig.  670)  is  expressed 
in  shade  line  merely,  without  hatching. 

Fig.  671  is  the  stub  end  of  a  coupling-rod.  The  bushes  are  solid,  of  brass, 
and  kept  from  turning  round  by  taper-pins,  which  are  secured  by  set-screws 
pressing  on  the  larger  end  ;  taper,  ^  in  3  inches. 

Fig.  672  represents  the  forked  end  of  a  cast-iron  connecting-rod  of  an  Eng- 
lish type,  the  end  of  the  working-beam  coming  within  the  forks.  Wrought- 
iron  connecting-rods  of  this  kind  are  most  generally  used.  One  side  of  the  fork 
is  shown  in  section,  with  its  bosses,  a  #,  and  the  cotters,  c  c. 


318 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


Fig.  673  is  a  section  of  the  lower  end  of  the  same  beam. 
The  lower  box  n  is  held  in  position  by  a  spherical  boss,  fill- 
ing a  recess  in  the  rod,  the  upper  brass  by  the  cotter ;  there 


FIG.  671. 

is  a  cover  c  over  the  box  and  crank-pin.  The  small 
channel  in  the  upper  box  is  for  the  introduction  of  oil. 
Cast-iron  connecting-rods  are  now  very  seldom  used. 
In  some  cases  of  vertical-beam  pumping  engines,  it  is 
necessary  that  the  water-load  of  the  pump  should  be 
counterbalanced  by  some  dead  weight  of  material,  and 
it  is  then  sometimes  convenient  to  make  use  of  a  heavy 
pump-connection.  The  wrought-iron  crank  connec- 


FIG.  672. 


FIG.  673. 


tions  of  American  river-boat  engines  are  peculiar  in  their 
construction.  They  are  made  as  light  as  possible,  with 
very  great  stiffness.  Fig.  674  represents  the  side  ele- 


FIG.  674. 


MACHINE  DESIGN   AND  MECHANICAL  CONSTRUCTIONS. 


319 


vation  of  such  a  connecting-rod.  The  means  adopted  to  give  the  required 
stiffness  consist  of  a  double-truss  brace,  a  a,  of  round  iron,  which  is  fastened 
by  bolts  to  the  rod  near  each  end  ;  struts  b  #,  cut  with  a  screw,  and  furnished 
with  nuts,  pass  through  the  center  of  the  brace,  by  which  means  the  braces 
are  tightened.  The  connecting-rod  at  its  smallest  part  near  the  extremities  is 


Fia.  675. 


FIG.  676. 


FIG.  679. 


of  the  same  diameter  as  the  piston- 
rod  ;  the  boss  in  the  center  is  from 
one  to  two  inches  more. 

Fig.  675  is  the  front  view  of  the 
forked  end  of  the  rod,  which  is  fitted 
with  the  usual  straps,  gibs,  and  cotters.  Fig.  676  is  the  side  view  of  the  brace-rod. 

The  cross-head  is  the  link  between  the  piston-rod  of  the  steam-engine  and 
the  connecting-rod  to  the  crank. 

Figs.  677,  678  and  679  represent  the  plan,  end  view,  and  section  of  the 
cross-head  adopted  by  the  Southwark  Foundry  for  their  high-speed  engines. 
It  is  of  cast-iron,  with  large,  flat  faces,  the  pin  p  for  the  connecting-rod 
being  in  the  middle  of  the  length.  This  pin  is  of  wrought-iron,  large  and 
flattened  on  top  and  bottom,  so  that  the  boxes  of  the  rod  can  never  bind  on 
the  pin  at  the  extreme  of  the  vibrations  of  the  rod  ;  usually  these  pins  are 
round.  The  pin  is  formed  with  large  squares  at  the  ends,  by  which  it  is  fitted 
into  the  jaws  of  the  cross-head,  where  it  is  secured  by  a  steel  pin  passing 
through  the  cross-head.  The  bearing  surfaces  of  the  head  and  those  of  the 
guide-bars  are  finished  by  scraping  to  true  planes  ;  there  are  no  means  of  ad- 
justment, as  there  is  no  wear  if  kept  clean. 

It  is  to  be  understood  that  the  piston-rod  moves  in  a  straight  line,  and  that 
the  stress  on  the  connecting-rod  pin  is  mostly  oblique.  Guides  are  to  be  pro- 
vided, between  which  the  cross-head  slides,  to  take  the  oblique  stress  off  the 
piston-rod. 

Figs.  680  and  681  are  elevation  and  plan  of  guide-bars  which  are  in  common 
use  for  both  vertical  and  horizontal  engines.  Lugs  or  ears  are  cast  on  the  steam- 
cylinders,  and  on  the  frames  to  which  the  bars  are  bolted,  and  between  which  the 
cross-head  slides.  The  grooves  or  notches  across  the  guide-bars,  at  the  ends  of 
the  stroke,  are  to  throw  off  any  grease  or  dirt  that  may  be  carried  along  by  the 
head  and  prevent  their  accumulation.  The  stress  on  the  guide-bars  is  due  to 
the  pressure  of  the  steam  on  the  piston  acting  obliquely  on  the  crank  through 


320 


MACHINE   DESIGN   AND  MECHANICAL  CONSTRUCTIONS. 


the  connecting-rod,  and  is  the  greatest  when  the  crank  is  at  right  angles  to  the 
piston.  It  can  be  determined  by  multiplying  the  pressure  on  the  piston  by 
the  length  of  the  crank,  and  dividing  the  product  by  the  length  of  the  con- 
necting-rod, which  will  be  the  stress  tending  to  separate  the  guides.  If  the 


FIG.  680. 


31 


FIG.  681. 

connecting-rod  be  3  times  the  stroke,  or  6  times  that  of  the  crank,  which  is  the 
usual  proportion,  then  the  stress  is  -J-  the  pressure  on  the  piston.  Sometimes 
the  proportion  of  connecting-rod  to  stroke  is  2%  to  1.  When  a  portion  of  the 
force  of  the  steam  is  opposed  directly  to  the  resistance,  as  in  direct-acting 
pumps,  and  only  the  irregularities  in  the  steam-pressure  are  transmitted  through 
the  connecting-rods,  the  proportion  of  rod  to  stroke  may  be  still  smaller.  In 
this  case  the  force  transmitted  to  the  fly-wheel  is  retransmitted  to  the  cross- 


FIG.  682. 


head,  whenever  the  resistance  in  the  pumps  exceeds  the  pressure  of  the  steam, 
thus  utilizing  the  expansive  properties  of  the  steam  by  a  cut-off. 

When  the  top  of  the  engine-frame  is  horizontal  it  may  form  the  lower 
guide  of  the  cross-head.     In  many  engines  the  guides  are  formed  in  the  frame 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


321 


itself  (Fig.  682),  in  which  the  bearing  surfaces  of  the  guides  are  arcs  of  circles 
within  a  pipe,  open  on  the  face,  which  forms  a  part  of  the  frame  and  is  bored 
at  the  same  time  with  the  cylinder,  and  conse- 
quently in  true  line.     On  locomotives  it  is  not 
unusual  to  have  the  guide  on  one  side,  as  in  Fig. 
683,  where  the  slide-bars  are  of  wrought-iron  and 
the  slide-block  is  fastened  between  the  two  plates 
of  the  cross-head  by  bolts.     It  is  the  most  com- 
mon practice  in  this  country  to  use  guides  with 
vertical  engines,  even  when  the  connection  is  with  **»•  688« 

working-beams,  but  abroad  the  parallel  motion  is 

more  popular.     The  working-beam  is  seldom  applied  to  stationary  engines,  but 
only  to  marine  and  pumping  engines. 


FIG.  684. 


Fig.  684,  elevation  of  engine  of  the  "  New  World,"  may  be  taken  as  the 
type  of  a  North  Kiver  steamboat  engine.     The  frame-work  is  composed  of  four 


21 


322 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


pieces  of  heavy  pine  timber,  d  d,  which  are  formed  into  two  triangles,  and  in- 
clined slightly  laterally  to  each  other ;  their  lower  ends  rest  on  the  keelsons, 
and  upon  their  upper  extremities  are  placed  the  pillow-block  c  of  the  work- 
ing-beam. They  are  solidly  fastened  together  and  to  the  boat  by  numer- 
ous horizontal  and  diagonal  timbers,  which  are  secured  by  wooden  knees  and 
keys,  and  are  heavily  bolted.  The  two  front  legs  are  bolted  to  flanges  cast  on 
the  sides  of  the  condenser,  and  the  other  end  of  the  framing  is  attached  to  a 
large  mass  of  timbers,  which  support  the  shaft  pillow-block  b  ;  the  framing  is 
further  steadied  by  two  additional  timbers,  and  rods  running  from  the  beam 
pillow-blocks  outside  the  shaft  to  the  keelsons  of  the  boat.  The  guides  a  are 
bolted  at  the  bottom  to  the  cylinder-flange,  and  retained  in  their  vertical  po- 
sition by  wrought-iron  braces  connected  with  the  framing.  The  height  of  the 
frame  is  46  feet,  width  at  bottom  31  feet. 

Figs.  685  and  686  are  views  of  the  working-beam  on  a  larger  scale.     It  is 
composed  of  a  skeleton  frame  of  cast-iron,  round  which  a  wrought-iron  strap 


FIG.  686. 

is  fitted  and  fastened.  This  strap  is  forged  in  one  piece,  and  its  extreme  ends 
are  formed  into  large  eyes,  which  are  bored  to  receive  the  end -pins  or  journals. 
The  skeleton  frame  is  a  single  casting,  and  contains  the  eyes  for  the  main  cen- 
ter and  air-pump  journals  ;  the  center  hub  is  strengthened  by  wrought-iron 
hoops  shrunk  upon  it.  At  the  points  of  contact  of  the  strap  and  skeleton,  key- 
beds  are  prepared.  Small  straps  connect  the  frame  and  main-strap  at  these 
points,  keyed  to  the  frame — keys  riveted  over.  The  frame  is  further  braced  by 
wrought-iron  straps,  C  C,  which  tie  the  middle  of  the  long  arms  to  the  ex- 
tremities of  the  shorter  ones.  The  following  are  the  general  dimensions  :  From 
center  to  center  of  end-journals,  26  feet ;  this  is  somewhat  less  than  the  usual 
proportion  to  length  of  stroke,  being  slightly  less  than  double  the  stroke  ; 
length  of  center  hub,  26",  a  a  ;  diameter  of  main  center  eye  c,  15f "  ;  of  air-pump 
journal-eye  d,  6f "  ;  of  end -journals  e  e,  8-J. 


MACHINE  DESIGN   AND  MECHANICAL  CONSTRUCTIONS. 


323 


Fig.  687  is  the  side  elevation,  Fig.  688  a  plan,  and  Fig.  689  a  section 
through  the  hub  of  a  cast-iron  working-beam.  The  proportions  are  as  in  prac- 
tice, but  the  end  as  shown  is  not  usual.  Fig.  690  shows  the  way  in  which  the 


FIG.  687. 


Fm.  688. 


connection-rod  is  attached,  the  dotted  lines  showing  the  head,  which  passes 
over  the  end  pivot.  The  common  form  of  the  end  is  like  that  of  the  working- 
beam  (Fig.  685). 


FIG.  690. 


From  the  following  table  of  practical  examples  from  "Architecture  of  Ma- 
chinery," it  is  safe  to  assume  as  a  rule  for  the  working-beams  of  land  engines, 
that  the  depth  at  center  should  be  the  diameter  of  the  cylinder,  and  the  length 
of  beam  three  times  the  length  of  stroke.  The  outline  is  parabolic,  having  for 
the  vertex  the  extremity  of  the  beam  and  the  point  B  in  the  curve  at  the  center. 
The  sectional  area  may  be  estimated  from  rules  already  given,  knowing  the 
load  at  the  extremity,  that  is,  the  pressure  on  the  piston,  the  weight  of  the 
same  and  its  connections,  and  also  the  force  required  to  drive  the  air-pump, 
estimated  at  the  extremity  of  the  lever.  As  an  engine  is  subject  to  shocks,  the 
load  should  be  estimated  at  six  times  the  absolute  load.  Five  per  cent  of  the 


324 


MACHINE   DESIGN   AND  MECHANICAL  CONSTRUCTIONS. 


nominal  power  of  the  engine  may  be  considered  the  maximum  of  power  required 
to  drive  the  air-pump. 


Diameter  of 
cylinder. 

Length  of 
stroke. 

Description  of 
work. 

Length  of  beam 
from  center. 

Depth  at 
center. 

Sectional 
area. 

Inches. 

Ft.   In. 

Ft.    In. 

Inches. 

Square  inches. 

tff 

8 

Rolling, 

12       4 

48 

240 

40f 

7 

Pumping, 

10       4 

36 

162 

39| 

6     9 

Blowing, 

9       6 

38i 

96± 

36f 

6     3 

Rolling, 

9       3 

30 

60 

24f 

5 

Mill-work, 

8 

25 

50 

18* 

4 

« 

6     10 

22^ 

50 

42 

4 

Marine, 

6       3 

23 

138 

42 

4     2 

u 

6       6 

27 

216 

32 

3 

II 

6 

22 

132 

Double  plates  or  flitches  of  wrought-iron  are  often  used  in  the  construction 
of  working-beams  and  side-levers.  Fig.  691  is  the  section  between  the  two 
plates  of  a  beam  of  this  kind,  attached  to  the  compound  pumping  engines  at 
Milwaukee,  Wis.  The  plates  are  each  30  feet  long,  by  6'  4"  deep  at  center,  by 
If"  thick.  The  connections  between  the  two,  shown  in  section  in  the  figure, 
are  cast-iron  pipes  with  wide  flanges  at  each  end  riveted  or  bolted  to  the  plates. 
The  main  center  and  other  small  journal-pins  are  rods  of  wrought-iron,  passing 
through  the  pipes,  and  extending  outside  the  plates  to  form  the  journals  ;  c  is 


FIG.  691. 


the  section  of  the  pin  for  crank  connection,  p  for  that  of  pump,  li  for  that  of 
high-pressure  cylinder,  I  for  that  of  low-pressure  cylinder,  m  for  main  center- 
pin,  and  g  for  the  parallel-motion  links.  This  last  is  usually  the  position  of 
the  air-pump  center,  but  in  this  engine  the  air-pump  is  below  the  high-pressure 
cylinder,  and  its  piston-rod  is  extended  to  the  air-pump  piston.  The  dimen- 
sions are— H.  P.,  36"  X  62"  ;  L.  P.,  58"  X  8  feet.  The  action  of  the  parallel 
motion,  in  keeping  the  cross-head  of  the  low-pressure  cylinder  in  a  vertical  line, 
will  be  understood  by  the  arcs  described  from  the  main  center  m  and  from  the 
fixed  point  a,  or  the  journal  of  the  radius  bar  #.  The  point  e,  the  angle  of  a 
parallelogram  formed  of  rods  and  links,  must  partake  of  the  motions  of  these 
two  arcs,  and  for  a  portion  of  movement  it  is  in  a  straight  line  parallel  to 
that  of  the  motion  of  the  piston-rod  cross-head.  It  is  usual  to  make  the 
radii  of  these  arcs  equal. 


MACHINE  DESIGN   AND  MECHANICAL   CONSTRUCTIONS. 


325 


Fig.  692  is  a  general  mode  of  finding  the  length  of  the  radius  rod  g  c.  F  is 
the  main  center  of  the  beam,  a  c  is  a  strap  or  link  attached  to  the  beam  at  a, 
the  piston-rod  to  be  attached  to  some  point  nearly  central  on  the  link,  which 
must  move  in  a  straight  line.  Moving  the  beam  up  and  down,  keep  the 
point  b  on  the  vertical  line,  and  mark  the  positions  of  the  lower  end  of  the 


FIG.  692. 


FIG.  693. 


link  c  c  c  ;  find  an  arc  which  will  pass  through  these  points,  and  the  center  of 
this  arc  will  be  the  fixed  center  g  of  the  radius  bar,  and  the  radius  that  of 
this  bar. 

Steam- Cylinders. — Fig.  693  is  a  sectional  plan  of  a  common  form  of  small 
steam-cylinder.  A  is  the  cylinder,  B  the  piston,  b  the  piston-rod,  D  the  slide- 
valves,  d  the  valve-rod,  C  the  valve-chest,  c  the  chest-cover,  s  s  the  steam-ports, 
e  the  exhaust-port,  S  the  stuffing-box  of  the  piston-rod,  s'  that  of  the  valve- 
rod.  H  is  the  front  head  and  H'  the  back  head  of  the  cylinder.  The  bolts 
attaching  the  heads  to  the  body  of  the  cylinder  are  not  shown. 

Length  of  Cylinder. — It  is  the  present  practice,  in  the  construction  of 
stationary  engines  for  driving  machinery,  to  make  the  stroke  not  over  twice 
the  diameter  of  the  cylinder,  and  for  diameters  above  24"  about  1^  times  the 
diameter  of  the  cylinder,  and  invariably  to  place  the  cylinders  horizontally 
with  a  direct  connection  with  the  crank,  without  the  intervention  of  a  work- 
ing-beam. 

Fig.  694  is  the  longitudinal  section  of  a  Corliss  steam-cylinder  which  has 
two  steam- valves,  s  s,  and  two  exhaust-valves,  e  e.  The  steam -pipe  S  is  at- 
tached to  the  top  of  the  steam-chest,  and  the  exhaust  E  to  the  bottom  of  the 
exhaust-channel ;  the  bolts  on  cylinder-heads  or  stuffing-box  are  not  shown. 
The  thickness  of  shell,  Mr.  Hawthorne  finds  by  many  examples  in  Corliss's 
large  practice,  to  conform  to  the  formula  t  =  *268  Vd ;  t  and  d  being  in 
inches.  Thus  the  thickness  of  the  shell  of  16"  cylinder  will  be  ^16  X  *268  = 
4  X  '268  =  1-072,  a  little  more  than  1".  The  thickness  of  flanges  should  ex- 
ceed that  of  the  shell  by  -J  to  £  its  thickness.  The  bolts  should  not  be  less 
than  y  and  seldom  more  than  1".  It  is  better  to  increase  the  number  of  bolts 
than  their  diameter,  the  breadth  of  flange  about  3  times  the  diameter  of  the 
bolts,  and  the  pitch  of  the  bolts,  or  the  distance  between  centers,  about  6  times 
the  diameter  of  the  bolts. 


326 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


Fig.  695  is  a  sectional  elevation  of  a  Cornish  pumping  engine's  steam-cylin- 
der. The  valves  are  in  pipes  outside  the  cylinder,  as  in  most  of  our  North 
Eiver  boats,  and  there  is  what  is  called  a  steam-jacket — that  is,  a  shell,  jj, 
outside  the  shell  of  the  main  cylinder,  inclosing  a  narrow  space  filled  with 
steam  by  a  pipe  connection  directly  from  the  boiler,  and  with  a  pipe  at  the 
bottom,  through  which  the  condensed  water  is  returned  either  directly  to  the 


boiler  or  discharged  into  the  hot  well.  All  steam-cylinders,  whether  with  or 
without  jackets,  should  be  clothed — that  is,  covered  with  some  preparation  to 
prevent  the  escape  of  heat  from  contact  with  air.  The  usual  clothing  is  hair- 
felt,  with  a  lagging,  ~b  I — that  is,  an  exterior  shell  of  some  wood,  usually  black- 
walnut. 

Figs.  696  and  697  represent  sections  of  two  types  of  water-cylinders.     In 
Fig.  696  the  pump-barrel  is  long  and  the  piston  short ;  in  Fig.  697  tho  pump- 


MACHINE  DESIGN   AND   MECHANICAL  CONSTRUCTIONS. 


327 


barrel  is  only  about  equal  to  the  diameter  of  the  piston  in  length,  but  the  length 
of  the  piston  is  equal  to  that  of  the  stroke  of  the  pump  and  that  of  the  pump- 
barrel.  The  figures  are  taken 
from  the  Worthington  pump,  and 
represent  his  arrangements  of 
valves  and  passage-ways.  1 1  are 


FIG.  696. 


FIG.  695. 

the  inlet  chambers,  i  i  the 
lower  valves,  and  o  o  the  upper 
ones.  A  is  the  air-chamber. 

Pistons  are  of  great  variety 
and  of  different  proportions, 
according  to  the  work  to  be 
done,  the  medium  in  which 
they  move,  and  the  friction 


FIG.  697. 


328 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


due  to  their  weight  on  the  sides  of  the 
cylinders. 

Fig.  698  is  the  cast-iron  piston  of 
a  locomotive.  The  spring  or  snap- 
rings  forming  the  packing  are  of  cast- 
iron,  1|"  wide  by  |"  thick,  of  uni- 
form section.  The  split  is  made  with 
a  half  lap,  and  the  splits  of  the  two 
rings  are  on  opposite  sides  of  the  pis- 
ton. The  outsides  of  the  rings  are 
turned  to  a  diameter  slightly  in  excess 

of  that  of  the  cylinder,  and  are  sprung  into  recesses  of  the  piston  fitted  to 
receive  them. 

Fig.  699  is  a  sectional  plan  and  Fig.  700  is  a  sectional  elevation  of  the  ex- 
terior of  a  piston-ring,  showing  another  common  form  of  ring  packing,  which 
consists  of  a  single  exterior  ring  r  and  two  exterior  rings  r"  r" ,  and  each  cut  in 

r" 


FIG.  698. 


FIG.  699. 


FIG.  700. 


two  and  so  fastened  that  the  joints  are  always  broken.  The  packing  is  set  out 
by  springs  s  s,  one  of  which  is  shown.  F  is  the  follower,  which  can  be  taken 
off  for  the  admission  of  the  rings  and  springs,  and  then  replaced  and  bolted 
to  the  piston,  making  a  close  joint  with  the  end  of  the  rings.  The  depth  of  the 
piston  at  the  exterior  is  from  3"  to  9",  varying  with  the  diameter  of  the  piston. 
Figs.  701,  702,  and  703  are  sections  of  the  exterior  rings  of  pistons  adapted 
more  particularly  to  water-pumps.  Fig.  701  depends  on  the  closeness  of  fit  of 


FIG.  701. 


FIG.  702. 


FIG.  703. 


the  exterior  of  the  piston  with  the  inner  surface  of  the  cylinder,  and  when  accu- 
rately turned  and  fitted  the  leak  is  very  inconsiderable.     By  the  use  of  grooves 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


329 


in  the  piston  (Fig.  702)  this  leak  is  still  further  reduced,  as  the  thread  of  the 
water  in  passing  through  the  joint  is  broken  by  the  grooves. 

In  Fig.  703  the  joint  between  the  piston  and  the  cylinder  is  made  tight  by 
a  gasket,  usually  of  hemp,  compressed  by  a  joint  ring  or  follower,  a,  in  the 
pocket  between  piston  and  cylinder. 

When  the  water-pressure  is  very  great,  as  in  hydraulic  presses,  peculiar 
packing-rings  of  leather  are  used. 


FIG.  704. 


FIG.  705. 


FIG.  706. 


Fig.  704  is  a  cup  leather  packing,  and  Fig.  705  is  a  U-packing.     The  ap- 
plication of  the  first  will  be  understood  from  Fig.  706,  in  which  the  piston  is 
packed  with  two  cup  leathers,  in  this  case  to 
withstand  pressure  in  both  directions.     Were 
the  piston  single-acting,  but  one  cup  would 
be  necessary — and  if  from  beneath  the  piston, 
this  would  be  the  lower  cup.      The  flexible 
flange  is  pressed  against  the   inside   of  the 
cylinder,  and  the  joint  is  perfectly  stanch. 

Fig.  707  shows  the  application  of  the  U- 
packing ;  it  is  put  into  a  recess  in  the  cylin- 
der by  bending  the  packing  into  a  saddle-bag 
form,  and  then  allowing  it  to  spring  back 
into  the  recess. 

Hemp  packings  are  made  to  serve  the 
same  purpose,  as  shown  in  Fig.  703.  They 
are  more  easily  made  than  the  U-packing,  but 
they  require  a  follower  or  cap  to  retain  them 
in  position. 

Packings  can  be  obtained  from  hydraulic- 
pump  and  press  manufacturers,  and  are  kept 
in  stock  of  all  the  usual  sizes.  Their  depths 
are  from  I"  to  1J"  for  diameters  varying  from  4"  to  14",  and  the  space  occu- 
pied by  the  thickness  in  the  U  from  f  to  f ".  A  filling  of  flat  braided  hemp 
is  placed  inside  the  IT  to  keep  it  tight  when  not  under  pressure.  The  pack- 
ings are  made  by  steeping  the  leather  in  warm  water,  forcing  them  into  a 
mold,  and  leaving  them  to  dry  and  harden.  The  molds  are  made  of  either 
metal  or  wood;  frequently  the  rings  are  of  metal,  and  the  piston  over  which 
the  cup  is  formed,  of  wood. 

Clearances  in  cylinders  include,  in  general  signification,  not  only  the  spaces 
between  the  piston  and  cylinder-heads  at  the  ends  of  the  stroke,  but  also  the 


FIG.  707. 


330 


MACHINE   DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


spaces  between  the  cylinder  and  the  valves  ;  and  as  those  spaces  are  voided  in 
a  steam-cylinder  at  each  stroke  for  which  adequate  work  from  the  steam  is  not 
obtained,  they  are  usually  made  as  small  as  possible.  If  the  steam  is  fairly  dry, 
from  y  to  1"  will  be  sufficient  for  end-clearances — that  is,  minimum  distance 
between  piston  and  cylinder-head. 

Piston-rods  are  proportioned  to  the  stress  on  them,  usually  one  square  inch 
of  section  to  each  5,000  pounds  of  stress.  In  Fig.  698  the  tapered  end  fits  a- 
taper  hole  in  the  piston,  and  is  riveted  over.  It  is  more  usually  held  by  a  nut, 
and  some  use  a  shoulder  on  the  inner  end  of  the  piston-rod  instead  of  a  taper, 
and  the  nut  brings  the  piston  strongly  up  against  this  shoulder. 

Piston-rods  are  made  either  of  steel  or  hammered  iron,  some  makers  of 
engines  preferring  one  and  some  the  other  material. 

Stuffing-boxes  are  the  mechanisms  to  prevent  the  leakage  of  steam,  air,  or 
water,  in  the  passage  of  the  piston  or  other  rod  out  of  the  cylinder  or  chest. 
They  consist  of  an  annular  chamber  around  the  rod,  filled  most  generally  with 
gaskets  of  hemp,  which  is  forced  down  by  a  ring  or  gland  into  close  contact 
with  the  rod  and  the  sides  of  the  box.  In  Fig.  693  there  are  two  stuffing- 
boxes  shown,  one  for  the  main  piston-rod,  the  other  for  the  valve-rod.  In  the 
latter  the  cap  of  the  gland  is  fitted  with  a  screw  to  connect  it  with  the  side  of 
the  stuffing-box,  by  which  the  gasket  may  be  more  or  less  compressed.  This 
is  the  general  form  of  stuffing-box  for  small  stems  or  pistons  used  on  steam- 
valves,  but  sometimes  with  a  ring  or  follower  on  the  top  of  the  gasket,  which 
is  forced  down  by  the  gland  without  turning  the  ring  or  gasket.  In  the  figure 

the  stuffing-box  is  made  of  brass,  and  screwed 
into  the  end  of  the  steam  or  valve-chest. 

The  stuffing-box  to  the   piston   is   cast 
with  the  head  of  the  cylinder,  and  is  bored 


FIG.  708. 


FIG.  709. 


out,  and  a  brass  bushing  fitted  and  driven  into  the  end  of  the  box.  The 
hole  through  the  bushing  in  most  boxes  fits  the  piston-rod  accurately.  The 
gland  is  of  cast-iron,  turned  to  fit  the  stuffing-box,  and  bored  to  fit  the 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS.  331 

piston-rod  ;  after  packing  the  box  the  gland  is  forced  in  and  retained  by 
screws. 

Fig.  708  is  the  plan  and  section  of  a  common  stuffing-box,  in  which  the 
thickness  of  packing  is  from  \"  to  1J",  and  the  depth  from  1£  to  2  times  the 
diameter  of  the  piston-rod.  The  number  of  bolts  vary  with  the  diameter  of 
the  piston — seldom  more  than  four,  and,  for  the  size  of  engines  mostly  in 
use,  but  two. 

Fig.  709  is  the  section  of  a  stuffing-box  of  the  proportions  adopted  by  the 
Southwark  Foundry.  Taking  the  diameter  of  piston-rod  A  as  the  unit,  B  is 
2,  C  3,  D  2,  all  scant  up  to  a  3"  rod,  or  22"  cylinder.  For  a  28"  X  42",  A  = 
4,  with  an  allowance ^of  £>"  for  clearance,  B  6f,  C  9,  D  6^". 

It  has  been  said  that  hemp  gaskets  were  in  most  common  use  for  the  pack- 
ing of  stuffing-boxes,  and  they  can  be  procured  readily ;  but  there  are  a  very 
great  variety  of  packings,  patented  or  otherwise,  which  are  very  good,  adapted 
to  common  stuffing-boxes  ;  and  there  are  also  metallic  packings  which  have 
given  great  satisfaction,  and  can  be  easily  procured. 

Valves — Steam- Cylinder  Valves. — The  simplest  and  most  common  is  the 
slide  D,  shown  in  Fig.  693.  The  function  of  the  valve  is  to  admit  the  steam 
alternately  into  the  ends  of  the  steam-cylinder,  and,  while  steam  is  being  ad- 
mitted through  one  port  to  one  end  of  the  cylinder,  the  other  end  is  being 
exhausted  or  the  steam  discharged  through  the  other  port. 
It  is  absolutely  necessary  (Figs.  710,  711,  712,  713)  that 
one  port  should  be  closed  before  the  other  is  opened,  that 
the  steam  may  not  be  admitted  to  both  ends  of  the  cylin- 
der at  the  same  time,  nor  that  it  may  flow  through  from 
either  end  into  the  exhaust.  The  simplest  form  of  valve 
is  shown  in  diiferent  positions  in  the  sections.  The  face 
of  the  valve-seat  is  shown  in  Fig.  713  ;  s  and  s'  are  the 
steam-ports,  and  e  the  exhaust-port.  The  valve  only  just 
covers  the  ports,  so  that  there  is  no  leak,  and  in  Fig.  712 
it  is  in  the  position  in  which  the  steam  can  neither  enter 
nor  escape  from  the  cylinder.  Suppose  there  be  a  move- 
ment of  the  valve  to  the  left,  the  steam  will  be  admitted  through  the  steam- 
port  s',  and  the  steam  can  escape  through  the  other  port  s  into  the  exhaust ; 
at  the  end  of  the  movement  of  the  valve  it  will  be  as  shown  in  Fig.  711,  with 
full  opening  of  steam  into  s',  and  full  exhaust  through  s.  If  the  motion  be  now 
alternated  the  ports  will  be  gradually  closed  till  the  valve  returns  to  its  first 
position  (Fig.  712),  and  then,  as  the  valve  continues  its  movement,  the  port  s 
begins  to  take  steam,  and  the  port  s'  to  connect  with  the  exhaust,  till  at  the 
end  of  the  motion  in  this  direction  the  valve  will  be  in  the  position  shown  in 
Fig.  710.  With  this  valve  there  can  be  no  economical  use  of  steam  ;  it  follows 
to  the  end  of  the  stroke  without  cut-off,  without  benefit  of  expansion,  except 
that  which  may  come  from  throttling,  that  is,  impeding  the  flow  resulting  from 
the  gradual  contraction  of  the  steam  openings. 

Of  the  Size  of  Ports  or  Openings. — Under  "Steam-pipes"  will  be  given 
the  formula  for  the  flow  of  steam,  but  the  general  rule  of  proportioning  the 
ports  of  a  cylinder  is  to  consider  the  velocity  of  steam  100  feet  per  second,  and 


332 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


of  the  exhaust  80  feet  per  second.  It  will  be  seen  from  the  movement  of  the 
slide-valve  that  the  opening  is  made  gradually,  and  closed  in  the  same  way, 
thereby  throttling  the  flow  of  the  steam.  To  avoid  this,  Mr.  Corliss,  in  his 
engine  (Fig.  694),  has  made  his  ports  long  and  narrow ;  the  steam-valves  open 
quickly  and  close  at  once  by  a  drop.  It  will  be  seen  that  the  valves  have 

cylindrical   faces   and   seats,    and   are   moved   by  a   central 

rocking-bar. 


From  the  great  size  of  the  common  slide-valve  in  pro- 
portion to  its  port,  the  bearing-surface  extending  all  round, 
there  ensues  a  great  pressure  on  the  surface,  tending  to  wear  it,  and  also  mak- 
ing the  movement  of  the  valve  more  difficult.  Various  expedients  have  been 
adopted  to  relieve  this  pressure,  which  is  especially  desirable  in  quick-running 
engines. 

Fig.  714  is  a  horizontal  section  of  cylinder,  through  steam  and  exhaust- 
valves,  of  a  Porter- Allen  engine,  and  Fig.  715  a  vertical  cross-section  through 
cylinder  and  valves.  The  valves  are  four  in  number,  one  for  admission  and  one 
for  exhaust,  at  each  end  of  the  cylinder,  and  on  opposite  sides.  They  stand 
vertically  so  as  to  drain  the  cylinder.  The  valves  work  between  opposite  par- 
allel seats  ;  the  exhaust-valves  nearly  and  the  admission-valves  wholly  in  equi- 
librium. The  action  of  the  back  plate,  and  how  the  wear  is  taken  up,  will  be 
understood  from  the  section  (Fig.  715),  which  passes  through  the  middle  of  one 
pressure-plate.  It  is  made  hollow,  and  most  of  the  steam  supplied  to  two  of 
the  openings  passes  through  it.  It  is  arched  to  resist  the  pressure  of  the  steam 
without  deflection.  It  rests  on  two  inclined  supports,  one  above  and  the  other 
below  the  valve.  These  inclines  are  so  steep  that  the  plate  will  move  down 
under  steam  pressure  ;  and  also  that  it  may  be  closed  up  to  the  valve  with  only 
a  small  vertical  movement,  the  pressure-plate  is  held  in  its  correct  position  by 


S 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


333 


projections  in  the  chest  on  one  side  and  tongues  projecting  from  the  cover  in 
the  other,,  which  bear  against  it  at  the  near  end,  as  shown.     Between  these 


FIG.  715. 


guides  it  is  capable  of  motion  up  and  down  and  back  and  forth  from  -fa"  to  -J". 
The  pressure  of  the  steam  on  this  plate  tends  to  force  it  down  the  inclines  to 


rm 


JTTL 


_(TTL 


rrn 


rest  on  the  valve.     By  the  means  of  the  screw  the  plate  is  forced  up  and  away 
from  the  valve,  and  can  be  so  nicely  adjusted  that  the  valve  works  freely  and 


334: 


MACHINE   DESIGN  AND   MECHANICAL   CONSTRUCTIONS. 


perfectly  steam-tight.     When  the  pressure  is  greater  in  the  cylinder  than  in  the 
chest,  the  pressure-plate  is  forced  back,  to  the  instant  relief  of  the  cylinder. 

Cylindrical  Valves. — Fig.  716  represents  the  section  of  the  steam-cylinder 
of  an  Armington  &  Sims'  steam-engine  with  a  cylindrical  valve.  The  steam- 
chest  S  is  central  and  incloses  the  valve  ;  the  exhaust  chambers  E  E  are  at  the 
ends  of  the  valve,  and  are  connected  through  the  hollow  stem  or  body  of  the 
valve.  The  valve  depends  on  its  accuracy  of  fit  for  its  tightness.  The  valve- 


FIG.  717. 


FIG.  718. 


chamber  is  bored  out  and  ground,  the  valve  is  turned,  ground,  and  carefully 
worked  by  hand,  to  so  close  a  fit  that  there  is  no  loss  of  steam  in  action,  and 
the  valve  is  completely  balanced. 

There  is  a  form  of  balanced  valves,  called  the  double-beat,  much  used  both 
for  steam  and  water  valves.  Fig.  717  is  a  sectional  elevation  of  a  steam  valve 
of  this  kind,  and  Fig.  718  a  plan  of  the  lower  seat  «,  with  the  valve-guides  g  g 
in  section.  There  are  two  seats,  a  and  ~b,  and  two  faces  on  the  valve  corre- 


sponding to  them.  The  balance  depends  upon  the  relative  diameters  of  the 
bearing-lines  of  the  two  faces.  In  the  figure,  if  the  exterior  of  the  bearing  at 
I  and  the  interior  at  a  are  both  tight,  the  valve  is  balanced  under  any  pressure, 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


335 


except  as  to  its  own  weight ;  s  is  the  valve-stem,  and  the  hole  r  is  for  a  bolt  to 
fasten  the  valve-seat  to  the  casting  of  the  steam-chest.     The  scale  is  -J  full  size. 

Fig.  719  is  another  form  of  balance,  consisting  of  two  equal  poppet-valves 
connected  together— the  steam  passage  to  the  cylinder  being  central,  and  the 
steam-chest  at  each  end,  and  connected. 

Automatic  valves,  that  are  moved  by  the  action  of  the  fluid  in  which  they  are 
placed.— Figs.  720  and  721  are  the  plan  and  section  of  a  disk  valve  for  the  air- 
pump  of  a  condensing  steam-engine.  The  valve 
consists  of  a  disk  of  rubber  lying  on  a  flat  grating 
or  perforated  plate  of  brass,  held  in  position  be- 
tween the  grating  and  a  spherical  guard  by  a 
-central  bolt.  The  shape  of  the  guard  gives  a 


FIG.  720. 


FIG.  721. 


uniform  flexure  to  the  rubber  in  lifting,  and  an  easy  flow  to  the  current  of  air 
and  water.  The  rubber  is  not  closely  clamped  between  the  guard  and  plate,  as 
will  be  seen  in  the  figure.  The  lower  nut,  after  being  screwed  home,  is  riveted, 
and  the  upper  nut  usually  pinned  to  prevent  turning.  The  size  of  the  apertures 
in  the  grating  are  adapted  to  the  thickness  of  the  rubber.  With  an  external 
diameter  of  opening  of  6",  and  rubber  -J-"  thick,  the  exterior  ring  of  openings  may 
be  f "  by  f",  the  lands  or  spaces  between  openings  J"  wide,  and  exterior  lap  of 
the  rubber  \  inch.  With  larger  diameters  and  larger  openings  thicker  rubber 
must  be  used.  This  valve  is  often  made  of  a  long  strip  or  flap  of  rubber,  on  a 
suitable  grating,  with  a  curved  guard  attached  on  one  side.  For  the  common 


air-pump  pressure,  £"  rubber  is  sufficient  for  apertures  1"  x  4".  With  the  use 
of  backing  and  face  plates  on  the  rubber  flaps,  the  gratings  may  be  dispensed 
with.  Thimbles  are  inserted  in  the  rubber,  and  the  rivets  connecting  the  two 
plates  pass  through  these  thimbles.  The  valves  to  the  Boston  sewage  pumping- 
engines  are  of  this  description.  Clear  opening  in  seats  13£"  x  4£",  rubber  £" 


336 


MACHINE  DESIGN   AND   MECHANICAL  CONSTRUCTIONS. 


SPACE  OCCUPIED  BY   THE 
VALVES. 


thick,  toe  of  guards  curved  to  a  2-£"  radius  for  the  hinge  of  the  rubber ;  the 
guards  have  leather  pads  for  the  valves  to  cushion  on  in  their  lift. 

Fig.  722  is  the  section  of  a  metallic  flap-valve  or  check-valve  of  the  Ludlow 
Valve  Manufacturing  Company  pattern.  Body  and  valve  are  of  cast-iron,  with 
valve  faces  and  seats  of  bronze.  The  bottom  of  the  case  B  is  flattened  and 
raised  toward  the  seat,  so  that  gravel  and  stones  may  not  lodge  against  it. 

In  Fig.  723,  section  of  a  like  valve,  there  is  a  small  secondary  valve  on  the 
exterior  of  the  main  valve,  which,  being  lighter  than  the  latter,  opens  earlier 
and  closes  later,  and  prevents  shock  to  the  main  and  to  the  valve. 

Check-valves  are  placed  outside  of  large  pumps  to  prevent  the  return  of 
water  in  cases  of  accident  to  the  pumps,  and  for  facility  of  their  examination. 

Valves  of  this  kind  open  from  the  pressure  of 
water  beneath,  and,  from  a  state  of  rest,  with 
some  suddenness  and  shock.  To  prevent  this  in 
large  valves,  there  is  a  valve  and  small  by-pass 
pipe,  from  one  side  to  the  other  of  the  valves,  by 
opening  which  the  pressure  an  the  two  sides  of 
the  valve  may  be  equalized,  and  the  excess  due 
to  the  starting  of  the  pump  distributed.  At 
many  pumping  works  the  by-pass  is  kept  open 
except  when  necessary  to  get  at  the  pumps.  In 
case  of  accident  to  the  pumps  the  flow  through 
the  by-pass  would  be  comparatively  small,  and 
readily  shut  oif. 

Fig.  724  is  a  section  of  a  poppet-valve ;  the- 
body  is  of  cast-iron,  but  the  valve  and  seat  are 
of  brass.  The  valve  is  guided  in  rising  and  falling  by  three  feathers  on  the 
valve.  The  lift  of  the  valve  is  controlled  by  the  projection  on  the  cover ;  a 
screw  is  often  substituted  for  this,  as  it  admits  of  adjustment  to  varied  lifts. 
Poppets  are  often  guided  by  stems. 

Fig.  725,  ball-valve,  guided  in  its  movement  by  an  open  cage,  c,  shown  in 


Measure- 

Measure- 

SIZE. 

ment  from 
face  to  face 

ment  from 
end  to  end 

of  flange. 

of  hub. 

Inches. 

4 

Hi 

13| 

5 

14* 

16 

6 

w* 

16 

8 

17| 

19 

10 

21| 

24 

12 

24! 

26-J 

16 

29 

31 

18 

33 

35 

20 

35i 

38 

24 

39f 

39 

FIG.  724. 


FIG.  725. 


section  and  attached  to  the  cover.     Ball-valves  are  usually  small  metallic  balls- 
on  metallic  or  wooden  seats,  or  rubber  balls  on  metallic  seats ;  and  cylindrical 


MACHINE  DESIGN  AND   MECHANICAL  CONSTRUCTIONS. 


337 


FIG.  726. 


yalves  have  been  made  of  the  same  section  as  in  the  figure  ;  the  body  of  the 
valves  of  brass  pipes  with  rubber  jackets. 

Fig.  726  is  a  section  of  a  rubber  disk-valve  in  very  common  use  in  direct- 
acting  pumps  and  small  pumping-engines  ;  sometimes  with  a  thimble  in  the 
rubber  as  a  guide  ;  usually  with  a  metallic  plate  on  top  of  the  rubber  for  the 
bearing  of  the  spring  ;  valve-seat  generally  of  composition, 
with  spindle  riveted  or  screwed  into  it.     Sometimes  the  rub- 
ber is  held  in  a  metallic  plate  or  cup. 

Large  valves,  either  poppets  or  disks,  are  objectionable 
from  the  great  lift  required  for  an  outlet,  proportionate  to  the 
area  of  opening  in  the  seat,  making  shocks  both  in  the  lifting 
and  seating.  Consequently,  these  kinds  of  valves  are  made 
small,  the  requisite  area  of  outlet  being  made  up  by  the  number  of  the  valves. 

The  balance-valve  (Fig.  717)  is  commonly  used  in  Cornish  and  large  pump- 
ing-engines.  From  its  two  beats,  the  lift  is  about  one  half  that  of  a  plain  valve. 
There  must  be  difference  enough  in  the  faces  to  admit  of  the  lift  of  the  valve 
by  the  pressure  of  water  acting  on  this  diiference.  The  seats  of  the  valves  are 
often  made  of  wood,  set  endways.  , Automatic  valves  should  have  springs  at 
their  backs  to  cushion  the  blow  on  the  lift,  and  to  start  the  valve  downward 
promptly  on  the  check  of  the  water-flow  at  the  end  of  the  stroke.  The 
great  desideratum  of  water-valves  is  that  there  should  be  little  lift  but  ample 
water-way. 

Valves  controlled  by  Hand. — Fig.  727  represents  a  side  view  of  a  water  bib- 
cock, called  a  /^ose-bib,  because  the  outlet  end  is  fitted  with  a  screw  to  adapt 
it  to  a  hose.     Without  this  screw  it  is  a  plain  Mb. 
If  both  ends  of  the  cock  are  in  the  same  line,  it  is 


FIG.  727. 


FIG.  728. 


FIG.  729. 


called  a  stop-cock.  The  ends  may  not  be  fitted  with  screws,  as  in  the  figure  ; 
the  screws  are  sometimes  female  screws,  and  often  with  taper  ends,  to  solder 
lead  pipe  to,  or  to  drive  into  a  cask.  These  cocks  come  under  the  common 
designation  of  plug-cocks,  from  their  interior  construction,  which  will  be 
readily  understood  from  the  section  given  in  Fig.  728.  They  are  used  in  both 
steam  and  water  pipes,  but  not  in  the  former  when  the  use  is  frequent  and 
daily,  and  then  usually  not  over  2"  in  diameter  of  passage. 

Fig.  729  is  the  side  view  of  a  compression  water-bib,  used  when  the  press- 
ure of  the  water  is  great.      The  section  is  somewhat  similar  to  that  of  Fig. 
732,  in  which  a  rubber  disk  is  forced  against  a  metallic  seat  to  shut  off  the  flow. 
22 


338  MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 

Figs.  730  and  731  are  side  views  of  common  air-cocks  for  boilers  and  steam 
work  ;  they  are  plugs  in  their  construction,  as  are  the  cocks  used  in  gas-fitting  ; 
size  of  air-cocks  to  f "  diameter. 

Fig.  732  is  the  section  in  part  of  a  globe  steam  or  water  valve  with  a  rubber 
disk  ;  soft  rubber  for  cold  water,  hard  rubber  for  hot  water  or  steam.  The 


FIG.  730.  FIG.  731. 


fluid  enters  below  the  diaphragm  ana  passes  up  through  the  aperture  in  it, 
which  is  controlled  by  the  valve  ;  a  screw  in  the  stem,  below  the  stuffing-box, 
bringing  it  in  close  contact  with  the  face,  or  raising  it  to  any  height  required. 


FIG.  732.  FIG.  733. 

They  are  called  globe-valves  from  the  shape  inclosing  the  valve  (Fig.  733). 
They  are  not  necessarily  rubber  disks ;  the  smaller  sizes  are  metallic  poppet- 
valves. 

The  dimensions  of  straightway  globe-valves  in  common  use  are  as  follows, 
from  "  Warming  Buildings  by  Steam  "  (Briggs)  : 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


339 


Diameter  of  open- 
ing in  seat. 

Body  —  gun-metal 
or  cast-iron. 

Nozzles  —  tipped 
or  flanged. 

Length  over  all. 

Diameter  of 
flanges. 

Number  of  bolts 
each  flange. 

Inches. 

Inches. 

1 

Gun-metal. 

M 

Tipped. 

u 

T45 

1-78 

i 

u 

u 

2-20 

f 

U 

u 

2-65 

1 

n 

u 

3-30 

1* 

a 

a 

3-85 

H 

u 

u 

4-35 

Cast-iron. 

(c 

6-10 

2 

Gun-metal. 

u 

5-30 

Cast-iron. 

U 

5-90 

u 

Flanged. 

5-75 

6 

4 

2* 

Gun-metal. 

Tipped. 

6'75 

Cast-iron. 

u 

7-30 

u 

Flanged. 

7-25 

7 

4 

3 

Gun-metal. 

Tipped. 

7'75 

Cast-iron. 

u 

9*25 

u 

Flanged. 

9-25 

n 

4 

8* 

a 

Tipped. 

10-25 

" 

Flanged. 

10-25 

8 

5 

4 

1 

u 

11-25 

9 

5 

5 

( 

u 

13-25 

10 

6 

6 

I 

U 

15-25 

11 

6 

8 

1 

»« 

19- 

13^ 

8 

10 

1 

II 

23- 

16 

10 

12 

1 

u 

27- 

19 

10 

Figs.  734  and  735  are  eleva- 
tions of  valves  of  the  same  type 
as  the  last,  but  from  their  form 
are  called  angle  and  cross  valves. 

Figs.  736  and  737  are  the 
plan  and  section  of  a  steam  valve 
of  the  Southwark  Foundry  pat- 
tern. Its  construction  and  action 
will  be  readily  understood  from 
the  drawing.  The  valve  is  with 
inclined  faces,  and  seat  ground 
to  a  fit,  and  is  guided  in  its 
movement  by  three  wings,  w,  w. 
This  is  a  common  type  of  throt-  FlG-  ?34.  Fio.  735. 

tie-valve  for  steam  use. 

It  will  be  observed  that  in  the  section  (Fig.  737),  and  especially  in  that  of 
the  globe-valve  (Fig.  732),  the  flow  of  the  fluid  passing  through  them  is  very 
disturbed  and  impeded  ;  to  avoid  this,  straightway  gates  are  almost  invariably 
used  on  water  mains,  in  which  the  gate  is  raised  entirely  out  of  the  line  of 
pipe,  so  as  to  leave  the  flow  unobstructed. 

Fig.  738  is  a  section  of  one  of  the  oldest  types  of  this  kind  of  valve,  the 
Coffin  valve,  with  double  disks,  d,  d,  self-adjusting  on  their  seats.  The  screw 
works  within  a  long  pipe  or  nut,  and  when  raised  the  disk-valves  are  above  the 
line  of  pipe  within  the  large  circular  chest. 

In  "  Scraps  "  is  a  perspective  view  of  a  similar  valve  of  another  maker. 


340  MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


341 


Fig.  740,  the  Safety-  Valve. — The  illustration  is  of  the  common  type ;  a 
poppet-valve,  with  a  stem  bearing  on  the  top,  and  this  weighted  by  a  scale- 
beam,  by  which  any  desirable  pressure  can  be  put  on 
the  valve.  To  every  boiler  it  is  absolutely  indispen- 
sable that  there  should  be  such  a  valve  attached  di- 
rectly, without  any  means  of  shutting  it  off,  as  in 

Fig.  740,  where  B  is 
the  boiler,  S  the  steam- 
pipe,  and  1)  the  blow- 
off  from  safety-valve. 
The  United  States  rules 
require  for  the  safety- 
valves  of  this  pattern, 


FIG.  Y?8. 


B 

FIG.  740 


for  ocean  and  river  service,  that  they  "shall  have  an  area  of  not  less  than  one 
square  inch  for  every  two  square  feet  of  grate-surface." 

"  But  when  safety-valves  are  used,  the  lift  of  which  will  give  an  effective 
area  of  one  half  of  that  due  the 
diameter  of  the  valve,  the  area  re- 
quired shall  not  be  less  than  one 
half  of  one  square  inch  to  two  feet 
of  grate-surface. " 

Fig.  741  is  what  is  termed  a  pop 
safety-valve  ;  the  steam  issuing  as 
the  valve  rises,  impinges  on  a  cup 
surface  to  force  the  valve  further 
open.  The  valve  is  held  down  by 
a  spring,  but  the  valve  can  be  raised 
by  the  lever  I.  Valves  of  this  kind 
are  often  inclosed  in  a  locked  box, 
that  they  may  not  be  tampered 
with. 

Hydrants. — For  water  -  service 
in  connection  with  high-pressure 
mains. 

Fig.  742  is  a  section  of  the 
Matthews  post-hydrant,  one  of  the 
best  known  of  the  type.  The  valve 
v  consists  of  a  series  of  leather  disks 
bolted  together  and  turned  coni-  FIG.  741. 


342 


MACHINE   DESIGN   AND  MECHANICAL  CONSTRUCTIONS. 


cal,  which  is  brought  in  contact  with  a  corresponding  seat  by  the  valve-rod 

and  its  screw  at  the  top  of  the  hydrant.      The  valve  is  opened  by  being 

forced  down  into  the  cavity  of  a  branch  of  the  pipe-main  ;  n  is  the  nozzle 

for  the  coupling  of  the  hose  ;  outside  the  main  pipe  of  the 

hydrant  there  is  a  case,  extending  from  near  the  line  of 

valve  to  the  ground  line,  called  the  hydrant  or  frost  case, 

which  prevents  the  hydrant  from  being  lifted  by  the  frost. 

Were  the  water  left  in  the  hydrant,  it  would  freeze  in  most 

exposures  during  winter ;   the  hydrant,  when  not  in  use, 

is  therefore  kept  empty.     This  is  effected  by  a  small  hole 

at  v,  which,  when  the  valve  is  closed,  is  opened,  and  the 

water  in  the  hydrant,  if  any,  is  discharged.     This  vent  is 

closed  by  a  slide  attached  to  the  valve-rod,  when  this  last 

is  moved  down  to  open  the  main  valve.     Instead  of  leather 

for  the  valve-face,  many  valves  are  fitted  with  rubber  ;  and 

there  is  also  a  great  variety  of  valves  for  hydrant  purposes — 

slides,  poppets,  disks — but  in  arrangement  of  hydrants  the 

illustration  is  almost   universally  followed;   often,  though, 

without  the  hydrant  case. 

Riveted  Joints,  as  used  in  the  construction  of  Boilers. — 
Tigs.  743-749  are  forms  of  rivets  with  their  proportions  re- 


FIG.  744. 


FIG.  749. 


FIG.  742. 


ferred  to  the  diameters  next  the  heads.  The  thickness  of  the  plate  connected 
by  rivets  will  be  given  in  a  table  hereafter.  Figs.  744  and  745  are  the  usual 
finish  of  rivets  in  hand-riveting  ;  Figs.  746  and  747,  when  done  by  machines. 
Fig.  748  is  a  counter-sunk  rivet,  the  head  being  flush  with  the  outside  of  the 
plate.  Fig.  749  is  the  head  of  a  rivet,  in  which  a  narrow  strip  at  the  edge  is 
burred  down  by  a  chisel,  or  calked,  to  make  the  joint  between  rivet  and  plate 
tight. 

Fig.  750  is  a  plan  and  section  of  a  single  riveted  lap-joint.     Joints  of  this 
kind  fail  from  the  tear  of  the  plate  on  the  line  of  rivets  if  the  rivets  are  too 


MACHINE  DESIGN   AND   MECHANICAL  CONSTKUCTIONS. 


343 


close  ;  by  the  shear  of  the  rivets  if  too  few ;  or  by  the  bursting  of  the  plate 
from  the  rivet  to  the  outside  if  the  space  is  too  small.  The  great  difference  in 
the  quality  of  boiler-plates  and  rivets,  and  the  uncertainty  as  to  the  effect  of 


O    -Q--S0- 

6   We- 


FIG.  751. 


punching  plates,  prevent  any  accurate  determination  of  the  exact  proportion 
of  riveted  joints.  We  insert  the  tables  from  a  practical  "  Treatise  on  High- 
Pressure  Steam-Boilers  "  by  William  M.  Barr.  Dimensions  in  inches  : 

TABLE  SHOWING  DIAMETER   AND  SPACING  OF  RIVETS  IN  SINGLE-EIVETED 

LAP-JOINTS. 


Thick- 
ness of 
plate. 

Diameter 
of  rivet. 

Length  of 
rivet. 

Center  of 
rivet  to 
edge  of 
plate. 

Center  to 
center  of 
rivets  or 
pitch. 

Thick- 
ness of 
plate. 

Diameter 
of  rivet. 

Length  of 
rivet 

Center  of 
rivet  to 
edge  of 
plate. 

Center  to 
center  of 
rivets  or 
pitch. 

A 

B 

C 

D 

A 

B 

C 

D 

A 

i 

1 

HI 

H 

i 

| 

2i 

If 

H 

i 

f 

li 

i 

14 

A 

| 

2| 

If 

2i 

A 

f 

H 

i 

1* 

f 

1 

24 

IT\ 

2f 

1 

f 

14 

1A 

2 

T¥ 

1 

3 

*A 

21 

A 

4 

2 

1A 

2| 

4 

H 

3i 

14 

3 

Single-riveted  joints  have  the  strength  of  about  56  per  cent  of  the  solid 
plate  ;  double-riveted  joints  about  70  per  cent.  Fig.  751  is  the  plan  and  sec- 
tion of  a  double  riveted  joint,  and  the  proportions  given  in  the  table  are  those 
recommended  by  Barr  : 

TABLE  SHOWING   DIAMETER  AND   SPACING    OF   EIVETING   IN    DOUBLE-RIVETED 

LAP-JOINTS. 


Thickness  of 


plate. 

Diameter. 

Length. 

Center  to  edge. 

Pitch. 

Center  to  center. 

Center  to  center 

0 

D 

E 

F 

i 

1 

li 

1 

2 

1* 

1A 

A 

f 

H 

1 

H 

2 

i» 

t 

f 

if 

1* 

B* 

H 

iff 

TV 

f 

2 

1* 

H 

2i 

H 

i 

1 

2i 

If 

3 

*& 

iff 

T9ir 

1 

H 

H 

3i 

*& 

2 

1 

1 

2| 

IA 

H 

2| 

2i 

H 

1 

3 

i* 

3| 

21 

2A 

1 

u 

3i 

if 

4 

3 

2i 

344 


MACHINE   DESIGN  AND   MECHANICAL   CONSTRUCTIONS. 


For  the  most  part  in  this  country,  rivet-holes  are  punched  ;  some  drill  them. 
By  punching  first  a  small  central  hole,  and  then  using  a  pin  or  teat-drill,  an 
annular  washer  is  taken  out,  leaving  a  clean  hole,  and  a  ready  means  of  test- 
ing the  quality  of  the  material  by  bursting  the  washer  by  a  drift.  It  is  the 
practice  here  to  make  no  boilers  less  than  £"  thick,  and  beyond  this  to  use  a 

factor  of  safety  of  six,  as  shown  in 
the  table. 

The  stress  on  the  circumferential 
seams  of  a  boiler  is  the  circular  or 
end  area  in  square  inches  multiplied 
by  the  pressure  per  square  inch,  and 
this  is  to  be  met  by  the  circumferen- 
tial section  of  the  shell.  The  longi- 
tudinal stress  can  be  estimated  by 
multiplying  the  diameter  of  the  boiler 

in  inches  by  the  pressure  per  square  inch,  and  this  stress  is  to  be  resisted  by  one 
inch  in  length  on  each  side  of  the  boiler,  or  by  a  section  of  plate  2"  wide  by  its 
thickness,  and  with  a  proper  factor  for  riveted  joints. 

Fig.  752  is  the  plan  and  section  of  a  single-riveted  butt-joint,  and  Fig.  753 
the  same  of  a  double-riveted  one.     The  two  plates  are  brought  close  to  each 
other,  and  the  joint  is  made  by  a  cover, 
proportioned  in  the  pitch  of  the  rivets 


Strength  of  solid 

SAFE  WORKING   LOAD. 

plate—  pounds  per 

square  inch. 

Single-riveted. 

Double-riveted. 

50,000 

4,700 

5,800 

60,000 

5,600 

7,000 

70,000 

6,500 

8,200 

o 


o 


o  o 


j) 


FIG.  752. 


FIG.  753. 


and  distances  of  centers  from  edges  of  plates,  as  in  rules  above  given;  and 
although  this  form  of  joint  in  some  cases  is  convenient,  it  has  not  been  found 
practically  stronger  than  the  lap-joint. 

But  butt-joints  with  double  covers,  one  on  each  side  of  the  plates,  increase 
the  shearing  resistance  of  the  rivets,  so  that  rupture  always  takes  place  in  the 


FIG.  754. 


FIG.  755. 


plates  ;  and  as  these  can  not  bend,  and  there  is  considerable  frictional  resistance 
between  the  plates,  the  strength  of  the  joint  has  been  found  to  be  more  than 
that  due  to  the  net  section  of  the  plates  between  the  rivets. 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


345 


Fig.  754  is  a  plan  and  section  of  a  combined  lap  and  butt  joint.  The  pitch 
of  the  exterior  rows  is  double  that  of  the  central  one  ;  for  a  f "  plate,  4"  for  the 
former  and  2"  for  the  latter. 

Fig.  755  is  the  plan  and  section  of  a  butt-joint  when  the  cover  is  of  T-iron — 
a  not  uncommon  form  of  strengthening  flues  to  resist  collapse. 


cjj©© 

©  o 

(J-vS.NI 

\s_ss\ 


- 


*(//>A 
Lvvxl 


FIG.  756. 


FIG.  757. 


FIG.  758. 


Junction  of  more  than  two  plates,  shown  in  plans  and  sections  (Figs.  756, 
757,  and  758). — These  become  necessary  when  cross-joints  intersect  longitudi- 
nal ones.  At  these  joints  one  or  more  of  the  plates  are  thinned  or  drawn  out 
by  forging. 

Fig.  759  is  the  plan  and  section  of  an  angular  connection  of  plates  by  the 
means  of  angle-iron  ;  this  should  be  a  little  thicker  than  the  plates,  and  its 
width  four  times  the  diameter  of  the  rivets. 


FIG.  759. 


FIG.  760. 


FIG.  761. 


FIG.  762. 


Figs.  760,  761,  and  762  are  sections  of  angular  connections  by  flanging  the 
plates.  The  iron  should  be  good  and  the  curvature  easy  ;  inside  radius  at  least 
four  times  the  thickness  of  the  plates. 


FIG.  763. 


FIG.  764. 


FIG.  765. 


Figs.  763  and  764  are  sections  of  joints  of  cylinders  of  unequal  diameters, 
or  surfaces  not  in  line  with  each  other. 

Figs.  765,  766,  and  767  are  sections  of  fire-box  legs. 

In  all  connections  provisions  are  to  be  made  for  the  means  of  holding  the 
head  of  the  rivet,  and  for  riveting  and  for  calking  the  joints. 


346 


MACHINE   DESIGN  AND   MECHANICAL   CONSTRUCTIONS. 


Fig.  768  is  the  perspective  view  of  a  boiler  of  the  type  most  commonly  used 
when  the  fuel  is  anthracite,  and  often  also  when  bituminous,  called  the  horizon- 
tal  tubular.  The  proportions  of  the  boiler  vary  with  the  requirements  of  their 
position,  and  with  the  views  of  the  mechanical  engineer  or  maker  constructing 
them.  Those  in  most  extensive  use  are  with  shells  of  4  to  5  feet  inside  diame- 
ter and  3"  to  3£"  tubes,  14  to  16  feet  long.  The  line  of  the  top  of  the  upper 
tubes  is  usually  about  -^  of  the  diameter  of  the  boiler  above  its  center  ;  tubes 
arranged  in  vertical  rows,  with  distance  between  tubes  £  of  their  diameter.  In 
my  own  practice  I  have  kept  the  average  distance  the  same,  but  making  them 
farther  apart  at  the  top  row,  say  J  diameter,  and  the  lowest  J-  diameter,  so  that 
the  line  of  tubes  is  radial  instead  of  vertical. 


FIG. 


The  following  table  is  from  Barr,  showing  the  greatest  number  of  tubes 
which  should  be  put  in  a  given  head,  no  tube  to  come  nearer  to  the  shell  than 
2"  for  boilers  of  small  diameter,  2J"  for  medium,  and  3"  for  the  larger  series  : 


Diameters  of 
bodies  inside, 
in  inches. 

NUMBER  OF  TUBES  (outside  diameter). 

Sin. 

8Jtn. 

3^  in. 

3|  in. 

4  in. 

| 

4J  in.              5  in. 

36 

26 

23 

20 

19 

16 

12                 10 

•       40 

34 

34 

25 

23 

20 

14             14 

44 

48 

36 

32 

25 

25 

20             16 

48 

50 

38 

36 

30 

26 

21              18 

52 

57 

50 

48 

38 

32 

26             21 

56 

72 

57 

55 

48 

41 

32              23 

60 

80 

68 

62 

55 

46 

36              30 

i 

A  (Fig.  768)  is  the  man-hole,  to  enable  the  mechanic  to  get  into  the  boiler 
to  examine  it.     It  consists  of  a  cast-iron  frame,  bolted  to  the  shell  of  the  boilers 


MACHINE  DESIGN   AND   MECHANICAL  CONSTRUCTIONS. 


347 


with  an  elliptical  opening  usually  9"  X  15"  in  the  clear  ;  the  valve  laps  about 
1"  on  each  side.  In  closing  the  opening  the  valve  is  passed  down  into  the 
boiler,  and  is  brought  up  against  the  valve-seat,  where  it  is  held  by  its  stem 
passing  up  through  a  movable  yoke,  and  brought  up  tight  by  a  nut  and  screw. 
The  joint  is  made  with  a  gasket  or  with  sheet-rubber.  The  man-hole  is  often 
placed  in  one  of  the  boiler  heads.  B  is  the  hand-hole,  of  the  same  general  con- 
struction as  the  man-hole,  but  smaller,  to  enable  the  fireman  to  clean  the  boiler. 
Formerly  this  hand-hole  was  quite  small,  but  of  late  the  practice  is  to  make 
them  6"  X  8",  or  even  as  large  as  8"  X  12"  for  large  boilers — in  fact,  a  man- 
hole. There  should  be  a  hand-hole  in  the  other  end  of  the  boiler,  so  that  by 
taking  off  both  hand-holes  one  can  look  directly  through  the  boiler.  As  this 
hand-hole  is  exposed  to  the  flame  and  products  of  combustion,  it  is  well  to 
make  it  smaller  than  the  front  one,  say  3"  X  5";  III  are  lugs  by  which  the 
boiler  is  supported  on  brick- work.  It  will  be  observed  that  in  the  head  above  the 
tubes  there  are  rivet-heads,  and  also  in  the  sides  back  of  the  first  seams  at  each 
end.  These  are  for  the  attachment  of  diagonal  stays.  The  tubes  themselves 
serve  as  stays  in  the  lower  part  of  the  boiler,  but  above  the  flat  surface  needs 
something  to  prevent  the  head  from  moving  out  under  pressure.  The  stays 
are  made  of  round  or  flat  iron,  bolted  directly  to  the  shell,  the  round  part 
being  flattened,  and  connected  by  a  yoke  and  pin  to  a  crow-foot  or  piece  of 
angle-iron  attached  to  the  head.  The  stays  are  from  f  "  to  1^"  diameter  or 
equivalent  sections. 

BARK'S  PROPORTIONS  FOR  STAY-BOLTS  FOR  FLAT  SURFACES. 


CENTER  TO  CENTER  OF  STAY-BOLTS   IN   SQUARE  INCHES. 

Pressure  per 

square  inch. 

*"  plate. 

iV  plate. 

|"  plate. 

A"  Plate. 

*"  plate. 

60 

H 

*f 

n 

H 

9 

80 

4f 

6* 

6* 

n 

w 

100 

4* 

4f 

Bi 

6i 

7 

120 

8| 

4i 

5 

Bf 

H 

140 

3f 

4* 

±f 

5i 

6 

Eigs.  769  and  770  are  a  longitudinal  and  a  half  transverse  section  of  an 
anthracite-burning  locomotive  from  the  New  Jersey  Railroad,  which  illustrates 
the  stays  used  in  such  forms  of  boilers.  Water-spaces  are  4"  wide  in  front,  3" 
at  sides,  and  6"  at  rear ;  stay-bolts  in  water-spaces  J"  diameter,  4"  centers. 
The  crown-sheet  of  fire-box  is  supported  by  double  cast-iron  girders,  extending 
across  the  boiler,  ends  resting  on  the  inner  plates  of  fire-box,  and  also  sup- 
ported by  hangers  h  h  from  the  outer  shell,  and  the  inside  of  the  steam-drum. 
These  hangers  have  a  fork  at  one  end,  through  which  a  pin  is  passed  to  con- 
nect it  with  the  foot  riveted  to  the  boiler ;  the  other  end  passes  into  the  space 
between  the  double  girder,  and  they  are  pinioned  together.  The  crown-sheet 
is  held  by  bolts  passing  down  through  the  double  girder.  The  bottom  of  the 
water-space  is  made  with  a  wrought-iron  ring.  The  opening  for  the  door  is 
made  by  turning  a  flange  on  the  inside  plate,  to  which  a  plate  ring  is  riveted, 


348 


MACHINE  DESIGN   AND  MECHANICAL  CONSTRUCTIONS. 


MACHINE   DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


349 


ind  the  joint  with  the  outside  plate  is  made  by  a  ring  of  angle-iron.     The 

ooiler  has  130  2"  tubes,  and  26  8J"  tubes. 

Fig.  771  is  a  gusset  stay,  used  in  angles,  con- 
sisting of  a  triangular  plate  with  the  edges  flanged 
and  riveted  to  the  shell. 


FIG.  771. 


FIG.  772. 


FIG.  773. 


Flue  Boilers. — Where  bituminous  coal  is  used,  small  tubes  become  clogged 

with  soot ;   it  was  therefore  customary 
to  construct  boilers  with  larger  tubes 
or  flues  of  boiler-iron  riveted  together, 
which  sometimes  failed  from  collapse, 
their   resistance    being   uncertain,    due 
largely  to  the  defect  of  an  accurate  cir- 
cular section.     Mr.  Fairbairn  made  ex- 
periments on  the  resistance  of  tubes  to 
collapse,  but  it  has  been  demonstrated 
that  the  rule  does  not  apply  within  the 
limits  of  length  adapted  for  boiler-flues, 
and  it  may  be  considered  ample  to  make 
the  tubes  subject  to  outside  stress 
fifty   per  cent    thicker   than    for 
bursting,  especially  for  the  large 
drawn    tubes   now   made.      From 
Mr.  Fairbairn's  experiments  it  was 
considered  necessary  to  make  the 
joints  of  tubes  subject  to  collapse 
as  in  Figs.    772   and   773,   which 
may  be   useful   against  deteriora- 
tion of  force  in  riveted  boiler-flues, 
and   might  in  long  mains   be   of 
importance,    especially   if   of  the 
form  of  Fig.   773,  which,  besides 
strengthening  the   tube,  provides 
for  expansion. 

Fig.    774  is  a  section   of  the 
Shapley  boiler,    as   made  by  the 
Knowles   Steam  -  Pump 
Works  —  a    good   form 
of  upright  boiler,  with 
the  crown -sheet  simply 
FIG.  774.  stayed  and  well  covered 


350 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS. 


by  water.     It  is  an  admirable  illustration  for  the  draughtsman  of  how  a  boiler 
in  action  may  be  represented. 

The  usual  form  of  upright  boiler  consists  of  a  fire-box,  extending  a  little 
above  the  door,  and  tubes  extending  from  the  crown-sheet  to  the  top-head, 
over  which  there  is  a  bonnet  to  secure  the  smoke,  which  is  led  off  by  a  smoke- 
pipe.  These  are  very  convenient  forms  of  boilers  for  furnishing  small  power, 
as  they  occupy  comparatively  little  space.  They  are  not  as  economical  in  their 
combustion,  and  they  are  very  apt  to  prime — that  is,  take  up  water  with  the 
steam. 

The  common  vertical  boilers  are  from  2  feet  6"  to  4  feet  6"  outside  diameter 
of  shell,  with  water  space  in  legs  of  2%"  to  3"  ;  extreme  height  of  boiler  from  2 
to  2J  times  the  outside  diameter  of  fire-box  ;  tubes  from  2"  to  2-J-"  diameter, 
and  spaced  from  V  to  H"  apart.  Water-line  from  10"  to  15"  above  crown-sheet. 
On  account  of  small  ground-space,  vertical  boilers  are  popular  with  some 
makers,  and  are  made  with  varied  appliances  to  secure  good  evaporative  re- 
sults and  to  protect  the  upper  joints  of  the  tubes  from  being  overheated. 

There  is  supposed  to  be  a  proportion  between  the  tube  sectional  area  and 
the  grate-surface,  say  from  \  in  the  horizontal  to  ^  in  the  vertical ;  but  this 
rule  is  entirely  empirical,  as  the  length  of  the  tube  is  a  large  factor  in  the  dis- 
charge of  products  of  combustion  (see  Sturtevant  tables  in  appendix).     There 
is  also  a  proportion  of  grate  to  heating  surface ;   but 
only  the  same  class  of  boilers  can  be  compared  with 
each  other,  as  fire-box  surface — and  that  exposed  di- 
rectly to  the  flame — is  much  more  effective  than  that 
of  the  tubes,  and  the  products  of  combustion  escape  at 
much  different  temperatures  in  different  boilers. 

Pipe    Connections. — Fig.    775   is   the  section   of  a 
flanged  connection  of  a  cast-iron  pipe  of  the  most  usual 
form,  but  some  thicken  or  reinforce  the  pipe  a  little  for 
1"  to  2"  in  length  next  the  flange  ;  but  if  there  is  a 
good  fillet  in  the  angle  of  the  flange  it  is  unnecessary. 
The  proportions  of  flanges  to  the  thickness  of  the  pipe  at  the  joint  are 
given  below  : 

DIMENSIONS  OF  CAST-IRON  FLANGED   PIPE  TO   WITHSTAND  SAFELY  A  PRESS- 
URE  OF  ONE  HUNDRED   POUNDS   PER  SQUARE  INCH. 


FIG.  775. 


Diameter  of  pipe 

4 

6 

8 

SO 

12 

16 

20 

Thickness  of  pipe  

§ 

JL 

1 

9 

| 

| 

Number  of  bolts 

5 

6 

8 

10 

10 

14 

18 

Diameter  of  bolts  

A 

4 

| 

1 

| 

| 

1- 

The  flanges  are  almost  invariably  faced,  and  joints  made  with  red  and  white 
,  or  a  sheet-rubber  washer.  Large  cast-iron  flanged  pipe  is  but  little  used 
for  street  mains  ;  water-service  socket-pipes  are  invariably  used,  and  for  steam 
connections  wrought-iron  pipe  is  to  be  preferred,  and  it  can  now  be  purchased 
of  any  necessary  diameter  ;  and  when  steam-drums  are  requisite,  or  very  large 
connections,  they  are  made  of  riveted  plate. 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


351 


Fig.  776  is  a  section  of  the  joint  used  by  Sir  William  Armstrong  for  the 
pipes  of  his  accumulator.  For  a  working  pressure  of  800  pounds  per  square 
inch,  pipes  of  5"  diameter  are  made  1"  thick  and  tested  to  3,000  pounds  per 
square  inch.  The  flange  is  elliptical,  and  there  are  but  two  bolts  ;  one  pipe 
slightly  enters  the  other,  forming  a  dovetailed  recess  in  which  is  placed  a  gutta- 
percha  ring  i"  diameter. 


FIG.  776. 


FIG.  777. 


FIG.  778. 


Figs.  777  and  778  are  sections  of  two  other  forms  of  cast-iron  flanged  pipes, 
both  with  projections  fitting  into  grooves.  The  packing  in  Fig.  778  is  a  ring 
of  lead.  In  Siemens's  air  reservoirs,  where  the  pressure  sustained  by  steel  rings 
is  1,000  pounds  per  square  inch,  the  joint  is  made  by  turning  a  V  groove  in 
the  face  of  the  rings,  and  placing  in  it  a  ring  of  annealed  copper  -fo'  diameter. 
This  form  is  adopted  by  many  mechanics  for  forming  flanged  joints  even  for 
steam  purposes. 

Wrought-Iron  Pipe  Connections. — With  the  present  cost  of  wrought-iron 
pipes  they  arc  almost  invariably  used  for  the  conveyance  of  steam,  but  are  more 
liable  to  rust  for  water  purposes  than 
cast-iron.  Wrought-iron  pipes  are 
either  butt-welded  or  lap-welded.  It 
is  a  mere  question  of  manufacture. 
It  is  difficult  to  make  a  lap-welded 
tube  less  than  1-J"  diameter,  and, 
therefore,  below  this  size  they  are 
usually  butt-welded  ;  but  this  size 
and  above,  lap-welded. 

Wrought-iron  pipes  in  continuous  length  are  connected  by  socket  or  sleeve 
couplings,  shown  partly  in  section  (Fig.  779),  which  are  almost  invariably  of 
wrought-iron.  A  thread  is  cut  on  each  end  of  the  pipes,  and 
internal  threads  in  the  coupling.  The  coupling  is  screwed  on  to 
the  end  of  one  pipe  and  the  other  pipe  screwed  into  the  coupling. 
The  screw  in  the  coupling  is  tapped  parallel  usually,  but  the  ends 
of  the  tubes  are  cut  with  a  taper  thread,  uniform  with  all  makers, 
of  1  in  32  to  the  axis.  The  length  of  the  screwed  portion  varies 
with  the  diameter. 

Fig.  780  is  the  longitudinal  section  of  tapering  tube-end  with  the  screw- 
thread  as  actually  formed,  and  considered  standard  by  the  late  Robert  Briggs, 
C.  E.,  in  his  "  Treatise  on  Warming  Buildings  by  Steam."  It  is  shown  in  the 
figure  double  full  size  for  a  nominal  2-J-"  tube. 


FIG.  779. 


352 


MACHINE  DESIGN   AND   MECHANICAL  CONSTEUCTIONS. 


FIG.  780. 
DIMENSIONS   OF  WROUGHT  TUBES  AND  COUPLINGS. 


DIAMETER   OF   TTTBE. 

CIRCUMFERENCE. 

Nomi- 
nal in- 
side. 

Actual  in- 
side. 

Actual  out- 
side. 

Inside. 

Outsid 

In. 

In. 

In. 

In. 

In. 

i 

0-27 

0-41 

0-85 

1-2^ 

i 

0-36 

0-54 

1-14 

1-7C 

|          0-49 

0°67 

1-55 

2-15 

i          0-62 

0-84 

1-96 

2-en 

f          0-82 

1-05 

2-59 

3'£( 

1             1-05 

1-31 

3-29- 

4'U 

li 

1-38 

1-66 

4  33 

5-21 

H           1  61 

1-90 

5-06 

5-9' 

2            2-07 

2-37 

6-49 

7'4( 

2i 

2-47 

2-87 

7-75 

9-0; 

3-07 

3-50 

9-64 

11-0( 

•i 

3-55 

4-00          11-15 

12-6* 

4 

4-03 

4-50          12-65 

14-1^ 

4* 

4-51 

5-00           14-15 

15-71 

5 

5-04 

5-56 

15-85 

17-4' 

6 

6-06 

6-62 

19-05 

20-8 

7 

7-02 

7-62 

22-06 

23-9, 

8     |       7-98 

8-62 

25-08 

27'K 

9 

9- 

9-69 

28-28 

30-4 

10 

10-02 

10-75 

31-47 

33-7 

Weight 
per  foot 
in  length. 

COUPLINGS. 

of 

Outside 
diameter. 

Length. 

Lbs. 

In. 

In. 

0'2t 

0-55 

*    ' 

0-42 

0-70 

1 

0-56 

0-83 

1 

0-84 

1-01 

1ft 

1-13 

1-24 

If 

1-67 

1-53 

If 

2-26 

1-89 

If 

2-69 

2-17 

2 

3-67 

2  68     !     2£ 

5-77 

3-19 

2f 

7-55 

3-87 

3 

9-06 

4  40 

H 

10-73 

4  99 

3J 

12-49 

5-49 

3f 

14-56 

6-19 

3i 

18-77 

7-24 

8* 

23-41 

8-36 

4 

28  35 

9-49 

4£ 

34-08 

10-54 

4£ 

40-64 

11-72 

5 

When  pipes  are  thus  put  together  in  lengths,  with  couplings,  it  is  frequently 
impossible  to  take  out  a  length  of  pipe  for  repairs  or  alterations  without  break- 
ing  a  coupling  or  fitting  ; 
provision  is  made  for  discon- 
nections by  the  insertion  of 
a  union  or  unions  in  the  line. 
Fig.    781   is  an  exterior 
view,  and  Fig.  782  a  section, 
of   the   common   malleable- 
iron  union  ;  p  and  p'  are  the 

•  781.  FIG.  782.  halves  into  which  the  tube 

is  screwed,  and  the  joint  is 

made  by  a  male  and  female  coupling.     The  male,  #,  turning  on  a  flange  on 
the  tube  p,  is  screwed  to  the  other  half  of  the  coupling,  and  the  joint  is  made 


MACHINE  DESIGN  AND  MECHANICAL   CONSTRUCTIONS. 


353 


tight  by  a  rubber  washer,  shown  in  black.  These  unions  are  used  only  in  the 
smaller  sizes  of  pipes.  The  flange  coupling  (Fig.  783)  is  preferred  by  most 
fitters,  and  they  are  made  of  diameters  up  to  14"  ;  the  thickness  is  about  one 
half  that  of  the  length  of  a  coupling  of  the  same  diameter.  The  bolts  are  from 
f"  to  f  ",  and  spaced  somewhat  larger  than  that  given  for  cast-iron  flanges. 
The  width  of  flange  is  such  as  to  admit  of  the  head  and  nut  of  the  bolt  without 
projection  beyond  the  edge  of  the  flange. 


FIG.  784. 


FIG.  785. 


FIG.  783. 


FIG.  786. 


Fig.  784  is  a  common  cast-iron  flange,  and  with  about  the  same  proportions 
as  in  Fig.  783.  When  the  lines  are  long,  and  provision  can  not  be  made  by 
bends  for  the  expansion  and  contraction  of  pipes  under  changes  of  temperature, 
a  fitting  like  a  stuffing-box  is  often  used,  the  end  of  one  of  the  tubes  being 
attached  to  the  box,  and  the  other  sliding  in  and  out  like  a  piston-rod  ;  some- 
times expansion  is  permitted  by  two  flexible  flanges,  admitting  of  a  sort  of  bel- 
lows-like movement ;  sometimes  by  a  connection  between  pipes  of  a  ring,  as 
in  Fig.  773,  or  a  succession  of  corrugations. 


FIG.  787. 


Fig.  785  is  a  soldering  union  ;  the  ring  b  is  like  that  of  the  male  coupling 
(Fig.  782),  which  is  screwed  directly  to  the  wrought-iron  pipe,  while  a  is  a 

23 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


brass  tube,  with  a  shoulder  on  the  bottom  on  which  the  coupling  a  turns,  and 
a  lead  pipe  is  soldered  to  the  tube.  If  it  is  not  necessary  to  break  the  joints,  a 
soldering  nipple  (Fig.  786)  only  is  necessary,  one  end  of  which  is  screwed  into 
the  wrought-iron  pipe,  and  the  other  soldered  to  the  lead  pipe. 

Figs.  787,  788,  789  are  taken  from  Briggs's  treatise,  and  give  the  dimen- 
sions of  the  parts  of  elbow,  tees,  crosses,  and  branches.  Fig.  788  shows  the 
parts  of  an  elbow  designated  by  letters  in  Fig.  787,  and  Fig.  789  shows  the 
applicability  of  the  same  to  tees  and  crosses.  The  scale  is  one  quarter  full 
size  ;  if  much  used,  it  would  be  better  for  the  draughtsman  to  construct  one 
of  full  size.  The  dimensions  are  obtained  by  measuring  from  the  base  or  zero 
to  the  inclined  lines,  on  ordinates  corresponding  to  the  inside  diameter  of  pipe 
required. 

Fig.  790  is  a  close  nipple  ;  Fig.  791  is  a  shoulder  nipple. 

If  the  uncut  part  of  the  tube  is  longer  than  in  the  figure,  it  is  called  a  long 
nipple  ;  they  serve  the  purpose  of  short  pipes. 


FIG.  792. 


FIG.  790. 


FIG.  791. 


Fig.  792  is  a  bushing.  There  is  a  thread  cut  inside.  It  is  screwed  into  a 
coupling,  and  the  pipe  that  is  screwed  into  the  bushing  must  be  smaller  in 
diameter  than  that  connected  with  the  coupling.  The  service  of  the  bushing 
is  to  connect  pipes  of  different  diameters,  but  the  reduction  of  one  side  or  arm 
of  a  coupling,  tee,  or  cross  is  better. 

Fig.  793  is  a  plug  to  close  up  the  end  of  a  pipe  by  screwing  it  into  the 
coupling  ;  caps  are  used  for  the  same  purpose  ;  half -couplings  with  one  end 
closed,  or  blank  flanges — that  is,  flanges  without  any  hole  through  them — 
bolted  to  a  flange  on  the  end  of  a  pipe. 

It  will  be  seen  by  Fig.  788  that  the  cast-iron  elbow 
makes  a  very  short  turn,  with  considerable  obstruction 
to  the  flow  of  the  fluid  through  it. 

Fig.  794  is  an  elbow  in  which  the  obstruction  is  very 
much  reduced.  The  angle  is  a  piece  of  wrought-iron 
pipe  curved  to  an  easy  radius  ;  and,  as  a  general  rule,  it 
may  be  said  that  for  the  connection  of  pipes  not  in  a  line 
with  each  other,  it  is  better  to  bend  the  pipe,  if  possible, 
than  make  angles  by  cast-iron  elbows. 

Figs.  795  and  796  are  a  tee  and  a  cross  as  used  in 
connections  of  hydraulic  presses,  made  of  composition. 
The  tubes  are  of  wrought-iron,  extra  thick.     The  usual  dimensions  for  such 
are  as  follows  : 


FIG.  794. 


MACHINE  DESIGN  AND  MECHANICAL  CONSTRUCTIONS.  355 


Outside  diameter. 
Inside  diameter . . 


r 

8>f 


I" 

]  II 


The  joints  are  made  by  leather  washers, 
square  ends  on  square  seats. 


FIG.  795. 


FIG.  796. 


FRAMES. 


Fig.  797  is  the  section  of  a  common  jack-screw,  in  which  the  pressure  is 
vertical ;  the  base  is  made  extended  to  give  it  stability. 


FIG.  797. 


FIG.  798. 


FIG.  799. 


FIG.  801. 


356 


MACHINE   DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


Figs.  798  and  799  are  side  and  end  views  of  a  cast-iron  housing  for  rolls. 
The  screw  exerts  the  pressure  on  the  box  of  the  roll-journal,  and  the  reaction 
is  a  tensile  pressure  on  the  sides  of  the  frame  ;  but  there  is  in  addition  much 
percussion  and  intermittent  stress  that  is  to  be  provided  against. 

Fig.  800  is  the  elevation  of  a  hydraulic  press,  and  Fig.  801  the  plan  of  top 
and  bottom  plates.  The  stress  on  the  rods  is  tensile,  and  they  must  be  of  sec- 
tional area  to  resist  securely  the  power  exerted  on  the  press.  The  plates  are 
beams  held  at  the  four  corners,  and  the 
stress  central.  The  platen  p  attached 
to  the  ram  is  braced  by  triangular  flanges 
from  the  hub.  The  cylinders  of  large 
hydraulic  presses  were  formerly  made 
of  cast-iron,  sometimes  hooped  with 


FIG.  802. 


FIG.  803. 


wrought-iron,  but  now  it  is  the  practice  to  make  them  of  cast-steel.  The  cyl- 
inders of  hydraulic  jacks  and  the  smaller  presses  are  made  of  drawn  steel  or 
wrought-iron  tubes. 

Fig.  802  represents  the  (side  elevation)  cam-punch  and  shear ;  in  this  case, 
the  force  exerted  while  the  machine  is  in  the  operation  of  punching  or  shearing 
tends  to  open  the  jaws  a  a  ;  and  the  tendency  increases  with  the  depth  of  the 
jaw,  the  stress  obviously  being  the  greatest  at  the  inmost  part  of  the  jaw.  The 
Irame  consists  of  a  plate  of  cast-iron,  with  two  webs  around  its  edges  ;  the  front 
web,  being  subjected  to  a  tensile  strain,  should  be  in  the  area  of  its  section  about 
six  times  that  of  the  rear  web,  which  is  subjected  to  a  compressive  force. 

Fig.  803  is  the  side-frame  of  a  planing-machine.  The  force  here  exerted  is 
horizontal  against  the  cutter,  which  can  be  raised  or  lowered  at  pleasure,  ac- 
cording to  the  magnitude  of  the  work  to  be  planed  ;  the  upright  has,  therefore, 
to  be  braced,  which  is  done  in  a  curved  form  for  beauty  of  outline. 

Fig.  684,  p,  is  a  wooden  frame  supporting  the  working-beam  and  shafts  of 
a  river-boat  engine. 

Fig.  682,  p,  is  a  side  elevation  of  a  horizontal  engine,  of  the  type  of  engine- 
frame  introduced  by  Mr.  Corliss  ;  (Fig.  804)  is  a  plan  of  the  same.  The  old 


MACHINE   DESIGN   AND   MECHANICAL  CONS 


[ONS. 


351 


type  of  steam-engine  frame  was  a  rectangular  cast-ipn  frame  ;  the  steam- 
cylinder  resting  on  the  top  side  flanges,  the  pillow-block  being  bolted  on  the  top 
of  one  side  flange,  and  the  crank  and  connecting-rod  forking  centrally  be- 
tween the  sides. 


FIG.  804. 


Fig.  805  is  a  side  view  of  the  inclined  wrought-iron  box-frame  of  the 
war  steamer  Susquehanna.  The  steam-cylinder  rests  between  the  frames,  and 
is  bolted  to  them.  The  two  frames  are  securely  stayed  to  each  other,  and 
bolted  to  the  keelson  and  the  bottom  of  the  ship.  For  small  inclined  engines, 
^  as  used  on  ferry-boats,  the  frames  are  of  wood, 

(  ^\.  as  also  in  many  of  the  horizontal  engines  of 

boats  on  Western  waters. 

Governors. — In  the  running   of  all 
machinery  there  are  variations  of 
speed,  due  to  varying  powers 
and  resistances,  caused  by 
increase  or  decrease  in 
the  pressure  pro- 
ducing the 


FIG.  805. 

power,  as  of  steam  or  water,  or  in  the  resistances  of  the  machinery,  from 
more  or  less  being  brought  into  action,  or  through  inequalities  of  work  done. 
To  maintain  the  speeds  at  as  much  uniformity  as  possible,  governors  are 
used,  which,  applied  to  steam-engines  or  water-wheels,  open  or  close  valves 
or  gates,  and  increase  or  reduce  the  supply  of  steam  or  water  to  the  cylin- 
ders or  wheels,  according  to  the  varying  necessities.  The  ordinary  gover- 
nor (Fig.  806)  consists  of  two  heavy  balls,  suspended  by  links  from  a  spindle, 
and  caused  to  revolve  by  some  connection  with  the  shaft  of  the  motor.  In  the 
figure  the  governor  is  driven  by  a  belt-connection  to  the  pulley,  p,  bevel-geared 
to  the  governor.  When  at  rest,  the  balls  hang  close  to  the  spindle,  but  when 
in  motion  the  balls  rise  by  the  centrifugal  force.  When  the  motor  is  running 
at  its  established  speed,  the  links  assume  a  position  nearly  at  45°  with  the 


358 


MACHINE   DESIGN   AND  MECHANICAL  CONSTRUCTIONS. 


spindle.  If  the  speed  falls  off,  the  balls  fall,  and,  acting  on  the  lever,  as 
shown  in  side  view,  open  the  valve  or  gate  controlling  the  passage  of  steam  to 
the  cylinder  or  water  to  the  wheel ;  if  the  speed  rises,  the  balls  rise  and  close 
the  valve  or  gate.  The  lever  does  not  always  connect  directly  with  the  gate, 
nor  is  there  always  a  lever,  but  the  rise  or  fall  of  the  balls  acts  on  some 
mechanism  which  performs  the  function  of  reducing  or  increasing  the  supply 
of  steam  or  water. 

The  size  of  the  balls  depends  somewhat  on  the  work  to  be  done,  the  resist- 
ance to  be  overcome  in  the  movement  of  the  gate  and  connections,  and  may  be 
much  reduced  if  this  work  is  thrown  on  some  other  mechanism,  which  is  usu- 
ally the  case  in  the  regulation  of  water-wheels ;  while  for  steam-engines  the 


FIG.  806. 


FIG.  807. 


work  to  be  done  by  the  governor  is  reduced  by  balancing  the  steam- valve,  or 
to  the  merely  setting  a  trip,  that  will  permit  the  movement  of  the  valve  at  any 
point  of  cut-off. 

In  the  Porter  governor  (Fig.  807)  the  balls  of  the  governor  are  compara- 
tively light,  but  they  are  connected  to  a  heavy  central  weight  by  levers,  the 
same  as  those  connecting  the  balls  with  the  spindle. 

Fly- Wheels. — In  most  machinery  there  would  be  great  inequality  of  move- 
ments, from  the  great  difference  in  power  exerted  or  resistances  to  be  overcome, 
and  in  the  application  of  the  force,  as  through  cranks.  To  obviate  this,  fly- 
wheels are  used,  which  absorb  energy  in  one  part  of  their  revolution  and  give 
it  out  at  another,  or  by  their  mass  in  movement  overcome  resistances,  as  in  the 
punching,  shearing,  and  rolling  of  metal,  which  comes  only  periodically,  and 
is  much  in  excess  of  that  usually  required.  In  addition,  fly-wheels  give  gov- 
ernors time  to  act,  and  consequently  the  motion  is  more  uniform  and  constant. 
For  the  speed  and  weight  of  fly-wheels  the  conditions  vary  so  much  at  differ- 
ent times,  even  with  the  same  engines,  that  it  is  impossible  to  get  data  for  an 


MACHINE  DESIGN   AND   MECHANICAL   CONSTRUCTIONS. 


359 


estimate  by  any  mathematical  formula  embracing  the  conditions.  From  the 
experience  of  the  best  mechanical  engineers,  and  from  published  examples  of 
constructions,  are  deduced  the  following  rules,  applicable  to  common  practice 
for  the  fly-wheels  of  steam-engines  :  The  diameter  of  fly-wheel  to  be  4  times 


FIG.  808. 


that  of  the  stroke  of  the  engine,  and  the  entire  weight  of  the  wheel  40  times 
the  square  root  of  the  diameter,  its  exterior  velocity  being  about  5,000  feet  per 
minute  ;  if  less  or  more,  increase  or  reduce  the  weight  inversely  as  the  veloci- 
ty. The  rim  is  generally  a  little  less  than  f  of  the  whole  weight.  For  rolling- 


360 


MACHINE  DESIGN  AND   MECHANICAL  CONSTRUCTIONS. 


mill  engines,  Mr.  C.  B.  Kichards  takes  the  weight  of  the  fly-wheel  at  60  times 
the  square  of  the  diameter  of  the  cylinder,  and  the  diameter  of  the  wheel  5 
times  that  of  the  stroke,  and  rim 
velocity  not  to  exceed  125  feet  per 
second. 

In  most  stationary  engines  the 
fly-wheel  is  a  pulley  or  band  wheel 
or  gear  driving  the  machinery,  but 


FIG.  810. 


uJ  LU  UJ   LU   '  Lf 


iTi  iTi    rfi  ffi  ffi 


FIG.  811. 

often  the  fly-wheel  is  independ- 
ent. Fig.  808  is  the  elevation 
and  section  of  such  a  wheel,  as 
built  by  the  Southwark  Foun- 
dry. The  construction  will  be 
understood  from  the  drawings, 
but  the  wrought-iron  links  con- 
necting the  segments,  shown  on 

a  larger  scale  (Fig.  809),  do  not  project,  but  are  counter-sunk  in  the  sides  of 
the  rim. 

Air-Chambers. — The  action  of  the  air-chamber  is  very  similar  to  that  of  a 


JJj 

JLLJ 

_____ 

n 

3D 

FIG.  812. 


MACHINE   DESIGN   AND   MECHANICAL   CONSTRUCTIONS.  361 

fly-wheel ;  it  tends  to  make  the  outflowing  or  inflowing  pressure  of  the  fluid 
uniform,  and  cushions  or  prevents  the  reaction  that  takes  place  from  the  fluid 
in  reciprocating  pumps,  especially  crank-pumps  ;  but  pumps  in  which  the  pis- 
tons or  plungers  start  very  slowly  and  stop  equally  so,  require  but  little  air- 
chamber.  Cornish  engines  are  usually  provided  with  a  stand-pump  instead  of 
an  air-chamber — that  is,  a  vertical  pipe  of  considerably  larger  diameter  than 
that  of  the  pump,  and  high  enough  to  contain  the  water-column. 

Fig.  810  is  the  section  of  a  copper  air-chamber  for  the  smaller  size  of  steam- 
pumps  or  hand-pumps.  It  is  screwed  into  the  top  of  the  pump-chamber. 

Fig.  811  is  the  elevation  of  an  air-chamber  for  power  pumps  of  larger  size. 
It  may  be  entirely  of  cast-iron,  or  a  cast-iron  base  with  a  copper  chamber.  A 
flange  is  cast  on  the  top  of  the  pump-chest,  and  the  chamber  is  bolted  to  it. 

Fig.  812  is  the  elevation  of  an  air-chamber  of  one  of  the  Brooklyn  pumping- 
engines. 

It  will  be  observed  that  the  lower  end  of  the  small  air-chamber  is  necked, 
or  of  smaller  diameter  than  the  main  part  of  the  chamber.  This  prevents  a 
too  sensitive  reaction  of  the  air  and  prevents  its  escape.  In  chambers  like 
that  of  the  Brooklyn  engine  it  is  good  practice,  for  the  same  purpose,  to  put 
a  diaphragm  across  the  inside  of  the  chamber,  perforated  with  holes.  When 
the  inlet  column  is  long,  whether  suction  or  under  pressure,  it  is  well  to  put 
an  air-chamber  on  it. 

Air-chambers  should  be  from  10  to  15  times  the  capacity  of  the  pump-cylin- 
der, with  glass  gauges  to  show  the  quantity  of  air  in  them  for  large  pumps, 
and  some  provision  to  supply  and  maintain  the  air  at  such  levels  as  will  be 
found  by  experiment  suited  to  the  easiest  working  of  the  pump. 


ENGINEERING  DRAWING. 


THERE  is  no  part  of  engineering  more  important  than  that  of  securing  a 
good  foundation  for  the  structure.  Where  likely  to  be  disturbed  by  frost,  the 
structure  should  start  below  it,  unless,  as  in  the  extreme  northern  regions  where 
frost  is  permanent  at  certain  depths,  the  support  should  be  in  it.  In  preparing 
the  foundation  for  any  structure,  there  are  two  sources  of  failure  which  must  be 
carefully  guarded  against  :  viz.,  inequality  of  settlement,  and  lateral  escape  of 
the  supporting  material  ;  and,  if  these  radical  defects  can  be  guarded  against, 
there  is  scarcely  any  situation  in  which  a  good  foundation  may  not  be  obtained. 
It  is  therefore  important  that,  previous  to  the  commencement  of  the  work, 
soundings  should  be  taken  to  ascertain  the  nature  of  the  soil  and  the  lay  of  the 
strata,  to  determine  the  kind  of  foundation  ;  and,  the  more  important  and 
weighty  the  superstructure,  the  more  careful  and  deeper  the  examination.  But 
it  must  be  understood  that  in  general  it  is  not  an  unyielding  but  a  uniformly 
yielding  stratum  that  is  required,  and  that  a  moderate  settlement  is  not  objec- 
tionable, but  an  inequality  of  settlement. 

In  good  sand  or  gravel,  the  common  load  per  square  foot  is  from  three  to 
five  tons.  Many  soils  are  very  compressible,  not  supporting  one  ton  per  square 
foot ;  if  the  structure  is  important,  the  bearing  resistance  of  the  strata  should 
be  tested  by  experiment.  The  base  of  the  wall  is  extended  to  secure  the  requi- 


FIG.  813. 


FIG.  814. 


FIG.  815. 


site  area  of  bearing-surface,  either  by  a  base-stone  (Fig.  813),  by  a  bed  of  con- 
crete (Fig.  814),  or  by  extending  the  wall  by  steps  (Fig.  815),  with  or  without 
concrete  base,  or  the  weight  may  be  distributed  by  inverted  arches  between 
walls  and  piers.  The  walls  themselves  should  sustain  from  three  to  ten  tons 
per  square  foot. 

When  the  foundation  is  beneath  water,  the  base  may  be  made  of  plank,  or  a 
grillage  of  plank  and  timber  (Fig.  816).  But  the  character  of  the  soil  must  be 
well  understood.  There  are  positions,  as  in  the  foundation  of  the  Custom- 


ENGINEERING  DRAWING. 


363 


House  and  other  public  buildings  at  New  Orleans,  in  which  it  would  appear 
that  there  could  be  no  practicable  area  of  surface  that  would  secure  a  perma- 
nent foundation  for  an  extensive  building.     A  pile 
foundation  in  such  earth  is  more  satisfactory,  but 
all  timber  should  be  covered  with  water  to  prevent 
rot. 

Figs.  817  and  818  represent  plan  and  elevation 
of  a  pile  foundation  ;  the  piles  are  usually  from 
10"  to  14"  diameter,  and  driven  at  about  3  feet  be- 
tween centers.  The  tops  are  cut  off  square,  and 
capped  with  timber  ;  the  caps  treenailed  or  rag- 
bolted  to  the  piles,  and  plank  spiked  to  the  timber. 
In  the  figure  a  sheet-piling,  s  s,  is  shown,  inclosing 

the  piles  ;  the  spaces  between  piles  and  timbers  are  often  filled  with  concrete, 
small  stone,  or  closely  packed  earth. 

Piles  are  used  either  as  posts  or  columns  driven  through  soft  earth  to  a  hard 
bottom,  or  depending  on  their  exterior  frictional  surface  to  give  the  necessary 
support,  either  in  earth  naturally  compact  or  made  so  by  the  driving  of  the  piles. 

In  the  first  case,  care  must  be  taken 
that  the  piles  be  driven  sufficiently 
deep  into  the  lower  strata  to  secure 
their  ends  from  slipping  laterally, 
and  soundings  should  be  made  care- 


FIG.  816. 


O      O 

O     O 


o  _o 
o  o 
o  o 
o  o 


fully  to  ascertain  the  dip  and  char- 


FIG.  817. 


FIG.  818. 


acter  of  these  strata.  In  many  places,  from  the  hardness  and  the  inclined 
position  of  the  lower  strata,  this  kind  of  foundation  is  inapplicable  and  unsafe. 

Where  a  firm  foundation  is  required  to  be  formed  in  a  situation  where  no 
firm  bottom  can  be  found  within  an  available  depth,  piles  are  driven,  to  con- 
solidate the  mass,  a  few  feet  apart  over  the  whole  area  of  the  foundation,  which 
is  surrounded  by  a  row  of  sheet-piling  to  prevent  the  escape  of  the  soil ;  the 
space  between  the  pile-heads  is  then  filled  to  the  depth  of  several  feet  with 
stones  or  concrete,  and  the  whole  is  covered  with  a  timber  platform  on  which 
to  commence  the  solid  work. 

In  the  case  in  which  the  support  from  the  piles  depends  on  the  exterior 
frictional  resistance,  the  rule  most  generally  adopted  by  engineers  is  that  of 
Major  Saunders,  published  in  the  "Journal  of  the  Franklin  Institute"  for  1851: 

Multiply  the  weight  of  the  ram  by  the  distance  which  the  ram  falls,  in 
inches,  at  last  blow,  divided  by  8  times  the  depth  driven  or  set  at  that  blow. 


364 


ENGINEEKING  DRAWING. 


Thus,  suppose  the  ram  to  be  1,600  pounds  weight,  the  fall  20  feet,  or  240",  and 

the  set  1  inch,  then  the  safe  load  would  be  —  -  =  48,000  pounds. 

1x8 

The  usual  weight  of  the  ram  or  hammer  employed  on  our  public  works  va- 
ries from  1,400  to  2,400  pounds,  and  the  height  of  leaders  or  fall  from  20  to  35 
feet ;  but  there  is  a  great  advantage  in  reducing  the  fall,  increasing  the  weight 
of  the  hammer,  and  the  frequency  of  the  blows.  As  generally  driven,  and  of 
average  size,  when  the  whole  weight  is  to  be  supported  by  the  pile,  ten  tons 
may  be  considered  a  usual  load,  but  when  additional  support  is  received  from 
compacted  earth,  broken  stone,  or  concrete  between  piles  and  caps,  this  bear- 
ing-surface should  also  be  taken  into  consideration.  In  some  loose,  sandy  soils 
piles  are  set,  not  by  driving,  but  by  the  water- jet ;  a  1"  or  1-J*  pipe  is  lashed  to 
the  whole  length  of  the  pipe,  and  a  force  of  water  through  this  pipe  clears  out 
a  hole  for  the  settlement  of  the  pile.  When  in  position,  the  pile  is  held,  the 
pipe  is  withdrawn,  and  the  sand  settles  around  the  pile. 

Iron  pipes  with  a  cast-iron  foot  are  sunk  also  by  a  water-jet,  the  water  being 
forced  into  the  pile  and  out  beneath  the  foot. 

Hollow  cast-iron  piles  have  been  driven  by  exhausting  the  air  from  the  in- 
side ;  then  the  weight  of  the  pile,  and  sometimes  an  added  load,  cause  the  pile 
to  settle  into  the  earth  ;  this  is  called  the  vacuum  process.  The  process  by 
plenum  is  by  expelling  the  water  out  of  the  pile  by  forcing  in  air  in  excess  of 
the  pressure  of  the  surrounding  water,  and  the  workmen  descending  within  the 
pile  and  excavating  the  material. 

Sheet-piling  (Figs.  819  and  820)  is  used  to  keep  water  out  from  a  foundation, 

or  to  prevent  the  passage  of  water 
through  the  earth,  as  in  an  em- 
bankment or  levee.  It  is  usually 
of  plank  two  to  three  inches  thick, 


FIG.  821. 


set  or  driven.     For  driving,  the 
bottom  of  the  plank  should   be 
sharpened  to  a  chisel-edge,  a  lit- 
FIG.  819.  FIG.  820.  tie  out  of  center  toward  the  tim- 

ber side,  and  cornered  slightly  at 
the  outer  edge,  that  it  may  hug  the  timber  and  the  plank  while  being  driven. 

Fig.  821  is  the  section  of  a  timber  sheet-piling,  in  which  a  tongue  and 
groove  forms  the  guide,  the  grooves  being  either  made  in  the  timber,  as  shown 
at  a  a,  or  planted  on,  b  b.  The  pile  should  be  of  uniform  thickness,  but  the 
widths  may  be  random  ;  six  inches  thick  is  a  good  practical  thickness,  driving 
well  under  short  and  frequent  blows  ;  the  tongue  should  be  of  hard,  straight- 
grained  wood,  2  inches  by  2  inches,  and  well  spiked  to  the  pile. 


ENGINEERING  DRAWING. 


365 


Frequently,  to  secure  the  foundation  from  water,  a  wall  is  constructed  of 
two  rows  of  sheet-piling,  driven  one  within  the  other,  and  the  space  between 
the  two  filled  with  clay  or  some  compact  earth.  This  is  called  a  coffer-dam;  the 
two  pilings  are  stayed  to  each 
other  by  bolts,  and  if  the  wall 
is  wide  enough  no  other  stays 
or  braces  will  be  necessary. 

Retaining '-walls  are  such  as 
sustain  a  lateral  pressure  from 
an  embankment  or  head  of  wa- 
ter (Figs.  822  and  823).  The 
width  of  a  re  tain  ing- wall  de- 
pends upon  the  height  of  the 
embankment  which  it  may  have 


to  sustain,  the  kind  of  earth  of 

which  it  is  composed  (the  steep- 

er  the  natural  slope  at  which 

the  earth  would  stand,  the  less  the  thrust  against  the  wall),  and  the  compara- 

tive weight  of  the  earth  and  of  the  masonry.     The  formula  given  by  Morin 

for  ordinary  earths  and  masonry  is  I  =  0  -285  Ji  -f-  h'  ;  that  is,  to  find  the  breadth 

of  a  wall  laid  in  mortar,  multiply  the  whole  height  of  the  embankment  above 


FIG  323 


the  footing  by 


1000 


for  dry  walls  make  the  thickness  one  fourth  more. 


Most  retaining-  walls  have  an  inclination  or  batter  to  the  face,  sometimes 
also  the  same  in  the  back,  but  offsets  (Fig.  822)  are  more  common.  The  usual 
batter  is  from  one  to  three  inches  horizontal  for  each  foot  vertical.  To  deter- 
mine the  thickness  of  a  wall  having  a  batter,  "determine  the  width  by  the 
rule  above,  and  make  this  width  at  one  ninth  of  the  height  above  the  base." 

Fig.  824  is  one  section  of  the  bulk-head  wall,  as  constructed  by  the  Depart- 
ment of  Docks  of  the  City  of  New  York,  on  the  North  Eiver  side,  and  in 
positions  where  the  mud  is  deep. 

The  site  of  the  wall  is  first  dredged  to  hard  mud  compacted  with  sand.  The  vertical 
piles  are  then  driven,  and  small  cobble-stones  mixed  with  coarse  gravel  put  around  and 
among  the  piles  to  the  height  of  the  under  side  of  the  binding  frames,  and  rip-rap  stone 
placed  outside  the  piles,  in  front  and  rear. 

The  binding  frames  are  then  slid  down  to  their  places.  These  binding  frames  were 
made  of  two  pieces  of  spruce  plank  5x10  inches,  placed  edgewise  one  over  the  other,  and 
running  from  front  to  rear  of  the  piles  between  the  rows.  An  oak  beam  8x8  inches  is 
let  through  these  planks  in  front  of  the  front  row  and  in  rear  of  the  rear  row  of  piles,  and 
an  oak  wedge  block  fitted  and  placed  by  the  divers  between  the  oak  beam  and  each  pile 
nearest  it.  The  duty  of  these  frames  is  to  hold  the  front  rows  of  piles  firmly,  in  case  there 
should  be  any  tendency  in  them  to  tilt  outward. 

More  cobble-stone  is  then  put  in  to  the  height  of  the  bottom  of  the  base  blocks  of  the 
\vall,  weighting  the  binding  frames  and  preventing  any  tendency  to  floating. 

The  bracing  piles  are  then  driven  on  a  slope  of  six  inches  horizontal  to  twelve  inches 
vertical,  between  the  rows  of  vertical  piles,  and  spaced  three  feet  from  center  to  center 
longitudinally  and  transversely.  All  the  piles  are  staylathed  and  adjusted  in  position  as 
soon  as  they  are  driven. 


366 


ENGINEERING  DRAWING. 


ENGINEERING  DRAWING. 


367 


The  bracing  piles  are  cut  off  at  right  angles  to  their  axis,  about  one  foot  below  mean 
low  water,  and  capped  with  twelve  inch  square  timber,  running  longitudinally.  The  sides 
of  the  caps  are  kept  horizontal  and  vertical,  and  a  sloping  recess  or  notch  made  to  receive 
the  head  of  each  bracing  pile,  and  give  it  a  good  bearing. 

The  six  rear  rows  of  vertical  piles  are  cut  off  at  two  inches  above  mean  low  water,  and 
notched  front  and  rear  to  give  an  eight  inch  wide  bearing  across  their  tops  for  the  trans- 
verse caps. 

The  three  front  rows  of  vertical  piles  are  cut  off  by  a  circular  saw,  suspended  in  the 
ways  of  a  pile-driver,  at  15*3  feet  below  mean  low  water  mark,  to  receive  the  concrete 
base-blocks  of  the  wall.  It  being  impossible  to  cut  off  piles  at  this  distance  below  the  sur- 
face of  the  water  to  exactly  the  same  height,  and  as  the  bottom  of  the  concrete  base-blocks 
would  rest  only  upon  the  highest  piles  of  those  under  them,  a  mattress  of  burlap,  con- 
taining freshly  mixed  soft  mortar,  in  a  layer  about  two  inches  thick,  placed  on  a  network 


FIG.  825. 


FIG.  826. 


of  marline  stuff,  supported  by  a  plank  frame  about  its  edges,  is  lowered  upon  the  tops  of 
these  piles  immediately  before  setting  the  base-blocks  upon  them.  The  diver  then  cuts  the 
netting  between  the  edge  of  the  mattress  and  the  plank  frame,  and  the  frame  floats  to 
the  surface  of  the  water. 


368 


ENGINEERING  DRAWING. 


The  base-block  is  then  immediately  placed  in  position  upon  the  mattress  of  mortar 
resting  on  the  piles,  and  the  excess  of  mortar  is  pressed  out  from  between  the  head  of  the 
pile  and  the  bottom  of  the  base-block,  until  each  pile  has  a  well  and  evenly  distributed 
portion  of  the  load  to  carry. 

The  concrete  base-blocks  for  this  section  are  T  feet  wide  at  the  bottom  and  5  feet  wide 
at  the  top  ;  on  the  front  the  vertical  height  is  13  feet,  and  on  the  rear  14  feet.  The  top 
has  a  step  on  the  rear  of  1  foot  height  and  1£  foot  wide,  extending  the  entire  length  of  the 
block,  for  the  purpose  of  giving  the  mass  concrete  backing  of  the  granite  superstructure  a 
good  hold  upon  the  block.  For  handling,  grooves  for  chains  are  molded  in  the  end,  and 
a  longitudinal  hole,  2  feet  in  the  clear  above  the  bottom,  connects  them,  with  the  corners 
rounded,  to  enable  the  chain  to  render  easily.  The  face  is  curved  inward,  to  save  material 
while  giving  a  broad  base ;  their  length  is  12  feet. 

After  the  blocks  are  set,  the  vertical  chain-grooves  in  each  block,  coming  opposite 
to  each  other,  are  filled  in  with  concrete  in  bags,  well  rammed  into  place.  This  closes 
the  joints  between  the  blocks,  and  also  acts  as  a  tongue  set  into  the  grooves  in  the 
blocks. 

As  soon  as  the  base-blocks  are  set,  and  the  groove  filled  in,  the  cross-caps  resting  on 
the  tops  of  the  vertical  piles,  and  on  the  longitudical  caps  of  the  bracing  piles,  reaching 
about  half  way  across  the  base-blocks,  are  placed  and  fastened.  Oak  treenails  are  used  in 
all  fastenings.  The  small  cobbles  are  then  filled  in  around  and  among  the  piles  to  the  top 
of  the  caps,  and  the  rip-rap  placed  in  the  rear  of  them. 

Figs.  825  and  826  are  the  elevation  and  plan  of  a  crib  with  dock  or  pier. 
Below  the  level  of  the  water,  as  here  shown,  the  logs  are  round  and  locked 
to  the  cross  timbers  ;  above  the  water  the  timber  is  squared,  the  exterior 
walls  presenting  a  tight,  smooth  surface  into  which  the  cross  timbers  are  dove- 
tailed. 


FIG.  827. 


Fig.  827  is  a  section  of  the  outer  wall  of  the  crib-pier  erected  on  the  West 
Bank  for  the  Quarantine  Department  of  the  Port  of  New  York  by  Mr.  J.  W. 


ENGINEERING  DRAWING.  369 

Ritch.     The  structure  consists  of  an  outer  wall  of  crib-work,  with  an  interior 
filling  of  sand,  228  feet  wide  by  488  feet  long.     The  interties  occur  at  inter- 
vals of  6  feet  spaces,  or  7  feet  centers. 
Extracts  from  specifications  : 

"  The  exterior  wall  to  be  built  in  blocks  up  to  low  water,  of  about  80  feet  in  length, 
sunk  to  a  line,  and  to  be  filled  up  to  low  water  with  stone-filling.  From  the  low  water 
the  construction  of  the  exterior  wall  will  be  continuous,  breaking  the  joints  of  the  logs 
throughout  the  entire  length.  The  base  of  the  blocks  will  be  formed  with  timbers  14  inches 
square ;  two  rows  on  the  outside,  held  together  with  interties  of  timber  12  inches  square, 
each  end  dovetailed  into  the  outside,  and  shiplapped  to  the  other  timbers,  secured  at  each 
end  and  intersection  with  iron  bolts,  1  inch  square,  14  inches  long,  well  driven  home. 

"The  cribs  of  the  entire  exterior  wall  to  be  built  with  sound  timber  12  inches  square, 
laid  so  that  they  touch  each  other,  secured  at  every  crossing  or  intersection,  and  in  the 
center  between  each  crossing,  with  iron  bolts,  1  inch  square,  20  inches  long.  The  cross 
timbers  to  be  all  in  one  length ;  the  ranging  timbers  to  be  in  lengths  of  not  less  than  46 
feet ;  joints  broken  over  the  logs  below.  The  cross-timbers  to  be  dovetailed  at  the  ends, 
and  shiplapped  at  intersections.  The  under  tier  of  timbers  to  be  secured  to  the  logs  below, 
the  ranging  timbers  to  the  under  tier,  and  the  upper  tier  to  the  ranging  timbers,  as  fol- 
lows :  at  each  end  and  every  crossing  with  an  iron  bolt,  1  inch  square,  21  inches  long, 
well  driven  home.  The  entire  exterior  to  be  close  fendered.  extending  from  the  deck- 
plank  to  low  water,  with  sawn  white-oak  plank,  5  inches  thick,  and  not  over  12  inches 
wide  ;  each  plank  to  be  secured  with  7  iron  bolts,  3  inches  square,  15  inches  long.  The  6 
corners  of  this  fendering  to  have  each  3  iron  bands,  5  feet  long  on  each  limb,  3f-  inches  by 
1  inch  counter-sunk  holes  to  receive  5  iron  bolts,  f  inch  square,  15  inches  long,  in  each 
limb. 

"Each  crib  to  be  filled,  from  the  floor-logs  to  within  6  inches  of  the  deck-plank,  with 
stone,  granite,  gneiss,  or  trap-rock  ;  none  of  the  stone  to  be  more  than  2  feet  in  any  direc- 
tion. The  entire  exterior  to  be  protected  with  stone,  in  large  pieces,  done  in  riprap." 

Fig.  828  is  a  transverse  section  of  the  river-wall  Thames  embankment,  Mid- 
dlesex side.  It  may  be  said  to  be  a  wall  of  concrete,  etc. ,  faced  with  granite, 
with  a  sewer  and  subway  within  the  same,  both  inclosed  by  brick-work.  In 
the  drawings  the  different  material  is  represented  by  different  shadings  and 
letters  :  g,  granite  ;  hi,  brickwork  ;  cc,  concrete. 

Extracts  from  specifications  : 

"The  embankment- wall  is  to  be  formed  within  iron  caissons  or  coffer-dams,  as  the  en- 
gineer may  direct.  As  soon  as  the  excavations  shall  have  been  made  to  the  requisite 
depths,  and  the  works  cleared  of  water,  the  trenches  shall  be  filled  up  with  concrete  to  a 
level  of  12|  feet  below  datum,  and  a  bed  dressed  to  the  proper  slope  and  level  for  the  foot- 
ings of  the  brick  wall.  This  wall  shall  be  formed  thereon  (when  the  concrete  has  become 
thoroughly  hard  and  consolidated)  at  a  true  slope  in  sets- off,  as  shown  on  drawing.  The 
brick-work  generally  shall  be  laid  in  courses  at  right  angles  to  the  face  of  the  wall.  The 
low  level  sewer  is  to  be  formed  on  concrete  foundation  carried  down  as  shown.  The  sewer 
shall  be  7  feet  9  inches  in  the  clear  diameter  for  a  length  of  1,820  feet,  and  8  feet  3  inches 
in  diameter  for  the  remainder  of  its  length,  the  whole  to  be  formed  in  brick-work  1  foot  11- 
inch  thick.  The  subway  shall  be  formed  7  feet  6  inches  high  by  9  feet  wide  in  the  clear, 
generally  ;  the  side-walls  to  be  18  inches,  the  arch  1  foot  H  inch  thick.  The  subway  sewer 
and  river-wall  shall  be  tied  into  each  other,  at  intervals  of  6  feet,  by  cross  or  counterfort 
walls  18  inches  thick,  extending  from  the  brickwork  of  the  wall  to  a  vertical  line  9  inches, 
beyond  the  side  of  the  sewer  farthest  from  the  said  wall,  and  from  footings  9  feet  belovr 
24 


370 


ENGINEERING  DRAWING. 


datum,  which  are  to  be  bedded  on  a  concrete  foundation  12  inches  thick,  up  to  the  under 

side  of  the  subway.     The  upper  arch  of  the  subway,  and  all  other  similar  arches,  shall  be 

coated  on  their  outside  circumference  with  a 

layer  of  Claridge's  patent  Seyssel  asphalt,  1 

inch  thick,  laid  on  hot,  and  returned  up  all 

spandrel  walls  rising  above  the  arch  to   a 

height  of  9  inches.     The  river-wall  shall  be 


faced  with  granite,  generally  to  a 
level  of  8  feet  below  datum,  and 
shall  be  surmounted  with  a  mold- 
ed parapet  of  solid  granite ;  the 
stones  to  be  laid  in  courses,  in  al- 
ternate headers  and  stretchers. 

"  The  beds  and  joints  to  be  full 
and  square  for  the  whole  depth,  so 
that,  when  set,  the  work  may  be 
close  and  solid  throughout ;  and 
no  joint  to  exceed  £  inch  in  thick- 


FIG.  828. 


ENGINEERING  DRAWING.  371 


ness.  The  whole  of  the  stones  above  the  given  level  (ll^-  feet  above  datum)  to  be  dow- 
eled together  in  bed  and  joints  with  slate-dowels,  not  less  than  5  for  every  foot  run  of 
wall  ;  each  2£  inches  square  at  least,  let  fully  2i  inches  into  each  stone,  very  accurately 
fitted,  and  run  in  with  neat  cement  ;  the  stones  to  be  bedded  and  jointed  in  cement,  and 
the  joints  struck  with  neat  cement. 

"The  whole  iron-work  to  be  delivered  on  the  works  perfectly  free  from  paint  or  other 
coatings." 

Fig.  829  is  an  isometrical  view  of  the  overflow  and  outlet  of  the  Victoria 
and  Regent  Street  sewers  in  the  Thames  embankment.  S  is  the  main  sewer, 
and  W  the  subway  shown  in  Fig.  828  ;  s  s  s  the  street-sewers,  discharging  into 
the  overflow  basin  0  ;  w  w  the  weirs  over  which  the  water  is  discharged  into 
the  weir-chamber  c  c  ;  p  is  the  penstock-chamber,  which  is  but  a  continuation 
of  the  weir-chamber.  It  has  been  attempted  in  the  drawing,  by  breaks,  to 
explain,  as  far  as  possible,  the  whole  construction.  Whenever,  from  storms, 
the  discharge  from  the  street-sewers  (s  s  s)  is  greater  than  can  be  carried  off  by 
the  main  sewer  (S),  the  water  rises  in  the  overflow-chamber  (0),  passes  over 
the  weirs  (w  w)  down  into  the  weir-chamber  (c),  then  into  the  penstock-cham- 
ber, and  through  the  flap-gates  (g)  into  the  river. 

Extracts  from  the  specifications  : 

"  The  foundation  to  be  of  concrete,  not  less  than  2  feet  in  thickness;  upon  this  brick- 
work shall  be  built  for  the  flooring  of  the  chambers,  and  for  the  side-end  and  weir-walls. 
The  weir-chamber  shall  be  divided  in  the  direction  of  its  length,  by  a  brick  wall,  into  two 
rectangular  overflow-channels,  covered  with  cast-iron  plates,  6  feet  8^  inches  long,  3  feet 
wide  by  •&  inch  general  thickness,  with  strong  ribs  and  flanges  on  the  under  side,  properly 
bolted  together  and  jointed  with  iron  cement,  and  bolted  down  to  stones  which  are  to  be 
built  into  the  under  side  of  the  brick-  work  of  the  basement-chamber.  Arches  on  either 
side,  running  parallel  thereto,  and  communicating  with  this  chamber  and  with  the  weirs 
which  are  to  be  formed,  upon  which  weir-walls,  divided  so  as  to  correspond  with  these 
.arches,  are  to  be  built  in  brick-work,  capped  with  granite  blocks,  4  feet  long,  2  feet  deep, 
and  2  feet  3  inches  in  the  bed.  The  floor  of  the  penstock-chamber  to  be  formed  with  York 
landings,  6  inches  thick,  having  a  fall  of  3  inches  to  the  river.  The  outlets  for  the  penstock- 
chamber  through  the  river-wall  shall  be  formed  by  an  arch-recess  in  granite,  and  fixed  with 
two  tidal  flaps,  well  hung,  and  firmly  secured  to  the  masonry  by  strong  bolts  and  screws. 

"  The  subway  is  to  be  continued  over  the  low-level  sewer,  and  across  the  overflow  cham- 
ber, by  cast-iron  plates,  curved  to  the  form  of  the  arch,  $  inch  general  thickness,  with 
strong  ribs  and  flanges  on  the  upper  side,  properly  bolted  together,  and  strongly  bolted 
down  to  the  brick-work  ;  jointed  with  iron  cement,  and  covered  with  brick-work,  to  form 
the  floor  of  the  subway.  From  a  point  of  10  feet  8  inches  on  either  side  of  the  central 
longitudinal  line  of  the  chamber,  where  the  sewer  and  subway  are  farthest  from  the  river- 
wall,  these  are  again  to  be  brought  into  their  general  position  by  two  curves,  each  not  less 
than  80  feet  in  length. 

"  The  whole  of  the  cast-iron  shall  receive  one  coat  priming  of  red  lead  and  linseed  oil, 
and  three  coats  best  coal-tar,  before  fixing  ;  and  the  accessible  surfaces  one  further  coat 
best  coal-tar,  when  fixed." 

Foundations  for  piers  and  abutments  of  bridges  beneath  the  surface  of  water 
are  formed  by  piles,  by  throwing  down  masses  of  stone  or  beton  until  the  mass 
reaches  the  surface  of  the  water,  by  open  caisson  or  by  inclosing  the  space 
within  a  coffer-dam,  and  proceeding  as  in  common  foundations,  or  by  an  in- 
verted caisson  and  air-lock. 


372 


ENGINEEKING  DEAWING. 


ENGINEERING  DRAWING. 


373 


An  open  caisson  is  a  chest  of  timber  which  is  floated  over  the  site  of  the 
work,  and,  being  kept  in  its  place,  is  loaded  with  stone  until  it  rests  firmly  on 
the  ground.  In  some  cases  the  stone  is  merely  thrown  in,  the  regular  masonry 
commencing  with  the  top  of  the  caisson,  which  is  sunk  a  little  below  the  level 
of  low  water,  so  that  the  whole  wood-work  may  be  always  covered,  and  the 
caisson  remains  as  part  of  the  structure.  In  others  the  masonry  is  built  on  the 
bottom  of  the  caisson,  and  when  the  work  reaches  the  level  of  the  water  the 
sides  of  the  caisson  are  removed. 

The  general  plan  adopted  by  G-.  A.  Parker,  C.  E.,  in  the  erection  of  the 
piers  of  the  Susquehanna  bridge,  was  : 

First  to  dredge  away  as  much  as  possible  of  the  material  in  the  bed  of  the  river  at  the 
pier  site.  A  f-inch  thick  boiler-iron  curb  was  then  sunk  and  secured  in  its  place.  The 
curb  was  about  30  feet  wide  and  50  to  60  feet  long,  and  of  sufficient  height  to  reach  above 
the  bed  of  the  river.  The  material  was  then  pumped  by  sand-pumps  out  of  the  curb, 
which  gradually  undermined,  and  settled  down  to  the  required  depth,  or  on  to  the  bed- 
rock. When  stumps,  logs,  or  bowlders  were  met  with,  they  were  removed  by  divers  work- 
ing in  a  bell.  After  the  rock  had  been  thoroughly  cleaned  off,  it  was  brought  to  a  uniform 
level  by  a  solid  bed  of  concrete  extending  over  a  greater  space  than  the  size  of  the  bottom 
of  the  pier,  using  the  diving-bell  for  this  purpose. 

Three  guide-piles  on  each  side,  and  one  at  each  end,  were  fixed  firmly  in  position.  A 
strong  platform  of  solid  timber,  the  size  of  the  bottom  of  the  pier,  was  then  placed  in 
position  over  the  curb,  and  at  the  surface  of  the  water.  On  this  was  placed  a  caisson  of 
iron  large  enough  to  contain  the  pier,  and  with  sides  and  ends  high  enough  to  reach  to  the 
level  of  high  water  after  the  caisson  is  landed  on  the  bottom.  The  caisson  was  then  made 
water-tight.  The  bottom  was  then  floored  over  with  masonry  and  stone,  and  laid  in  mor- 
tar up  the  sides  of  the  caisson  to  the  top,  thus  constituting  a  stone  caisson  inside  of  an  iron 
one.  This  was  secured  to  the  guide  piles,  and  the  masonry  of  the  pier  proper  was  laid  up, 
the  caisson  sinking  as  the  weight  of  masonry  inside  increased,  until  it  finally  settled  upon 
the  bottom  which  had  been  prepared  for  it,  as  already  described.  At  some  of  the  piers 


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FIG.  830.                                                                                    FIG.  831. 

(Figs.  830  and  831)  screw-rods  were  used  to  suspend  the  pier  and  gearing  attached,  gov- 
erned by  one  man,  who  at  pleasure  could  raise  or  lower  without  assistance  the  whole 
pier.  Wooden  piles  were  driven  and  cut  off  by  machinery  just  above  the  ground,  and  the 


374:  ENGINEERING  DRAWING. 

platform,  with  its  incumbent  pier,  lowered  upon  them  ;  at  other  piers  the  foundation  was 
on  rock. 

Piers  are  sometimes  made  by  sinking  a  wrought-iron  curb,  extending  from  the  bottom 
to  above  the  level  of  the  water,  driving  Within  it  the  usual  proportion  of  piles,  and  then 
filling  the  spaces  entirely  with  concrete. 

Dams  are  constructed  to  pond  water  for  the  supply  of  cities  and  towns  ;  for 
inland  navigation,  by  deepening  the  water  over  shoals,  and  the  feeding  of  ca- 
nals ;  for  power  in  its  application  to  mills  and  workshops  ;  and  for  irrigation. 
To  whatever  purpose  the  water  is  to  be  applied,  there  are  two  questions  to  be 
settled  :  Whether  the  level  will  be  raised  high  enough  by  the  construction,  and 
whether  the  flow  of  the  stream  be  sufficient  for  the  purpose  required  ;  and  fur- 
ther, it  may  often  be  important  to  know  how  large  a  pond  will  be  thus  formed, 
how  ample  a  reservoir  for  unequal  flow,  or  intermittent  use.  If  the  pond  be  small, 
so  that  the  water  can  not  be  retained,  and  the  supply  is  only  the  natural  run  of 
the  stream  at  a  high  level,  then  the  minimum  flow  of  the  stream  is  the  measure 
of  its  capacity. 

The  rule  that  obtains  on  the  Merrimack  River,  at  Lowell,  and  Lawrence, 
where  the  pondage  is  more  than  the  average,  is  that  1  cubic  foot  per  second 
per  day  of  12  hours  per  square  mile  of  water-shed  can  be  depended  on  for  per- 
manent mill-power.  On  very  small  streams  it  may  often  happen  that  pondage 
may  be  secured,  and  the  supply  be  equal  to  one  half  the  rain-fall. 

Blodgett,  in  his  "  Climatology  of  the  United  States,"  says  that  "  in  this 
sense  of  permanence  as  a  physical  fact,  we  may  consider  the  quantity  of  rain 
for  a  year  as  a  surface-stratum,  on  the  Atlantic  slope  and  in  the  central  States 
of  3|  feet,  which  may  be  diminished  to  half  this  quantity,  or  increased  to  twice 
as  great  a  depth  in  the  extreme  years.  But,  with  such  an  average  and  such  a 
known  range,  we  may  deal  with  the  quantity  as  definitely  as  with  a  stream  of 
which  we  know  the  mean  volume  and  the  extremes  to  which  it  is  liable,  and 
for  many  departments  of  engineering  these  climatological  measures  are  as  indis- 
pensable as  those  of  tide  or  river  hydrography." 

The  evaporation  from  a  reservoir-surface  at  Baltimore,  during  the  summer 
months,  was  assumed  by  Colonel  Abert  to  be  double  the  quantity  of  rain-fall. 
Dr.  Holyoke  assigns  the  annual  quantity  evaporated  at  Salem,  Mass.,  to  be 
56"  ;  but  from  experiments  made  by  the  Croton  Aqueduct  Department,  in 
1864,  of  the  evaporation  from  a  box  set  in  the  earth-bank,  and  two  afloat  in 
the  upper  reservoir,  the  quantity  was  found  to  be  severally  37*12,  37 '53,  and 
39*97  inches. 

Fig.  832  is  the  section  of  a  crib-dam  in  northeastern  Colorado  for  the  pond- 
age of  water  for  the  purposes  of  irrigation.  The  crib-work  is  of  round  logs, 
10"  at  least  in  diameter,  joined  at  the  ends  as  in  ordinary  log  huts,  with  dove- 
tail or  tongue.  Each  crib  is  18  feet  long  on  the  face,  and  the  fastenings  are 
2"  X  18"  treenails.  The  cribs  are  set  radially,  forming  a  curve  up-stream  of 
200  to  238  feet  radius.  The  crib  gives  the  stability,  but  the  water-tightness 
depends  on  a  shutter,  p,  or  vertical  panel  of  timber,  on  the  up-stream  side  of 
which  there  is  a  filling  of  earth. 

Crib  or  wooden  dams,  when  the  timber  is  not  kept  covered  with  water,  fail 
from  the  decay  or  rot  of  the  timber. 


ENGINEERING  DRAWING. 


375 


Fig.  833  is  a  section  of  the  dam  across  the  Croton  River,  constructed  under 
the  direction  of  Mr.  John  B.  Jervis,  for  the  supply  of  the  aqueduct  for  the 
city  of  New  York.  This  dam  was  built  on  an  earth  foundation,  with  curved 
roll  in  cut  stone,  extended  by  a  timber-apron  some  50  feet,  supported  by  strong 

P 


FIG. 


crib-work.  Originally  there  was  a  secondary  dam  still  farther  down,  to  throw 
back-water  on  this  apron.  In  the  erection  of  this  dam,  excavation  was  made 
of  all  loose  material ;  the  cribs  C  and  D  were  built  up,  and  the  tops  were 
planked  ;  on  this  planking  were  carried  up  the  cribs  F  and  G.  Between  these 
piers  the  space  E,  as  well  as  e  below  and  on  the  cribs,  was  filled  in  with  con- 
crete ;  on  this  the  body  of  the  dam  was  erected  in  stone-masonry,  laid  in  cement. 
The  face-work  of  granite  is  cut  to  admit  of  a  joint,  not  exceeding  ^  of  an  inch. 


FIG.  833. 


Above  the  dam  is  an  earth  embankment,  its  upper  part  protected  by  a  rubble- 
paving.  The  radius  of  the  granite  face  is  55  feet,  and  the  dam  38  feet  high 
from  level  of  apron  to  crest  of  dam. 

Fig.  834  is  a  section  of  the  dam  across  the  Connecticut  River,  at  Holyoke, 
Mass.     This  dam  is  1,017  feet  long  between  abutments,  and  averages  30  feet 


376 


ENGINEERING  DRAWING. 


high  by  a  base  of  80  feet.     It  is  constructed  of  tim- 
ber crib- work,  loaded  in  with  stone  for  about  ^  its 
height.     The  foot  of  each  rafter  is  bolted  to  the 
ledge,  and  all  timbers   at  their  intersections   are 
treenailed   together  with  2"  white-oak  treenails. 
The  inclined   plank-face  is  loaded  with  gravel, 
and  the  joint  at  the  ledge  covered  with  concrete. 
The  lower  or  base-tier  of  ranging  timbers  were 
15"  X  15"  ;  the  other  timbers,  12"  X  12".     The 
rafters  are  placed  vertically  over  each  other,  in 
bents  of  6  feet  between  centers.     The  plank- 
ing was  of  hemlock,  6"  thick,  with  oak  cross- 
planking  at  crest  of  dam,  4"  thick  at  bottom 
.and  8"  at  top.     The  crest  was  plated  with 
iron,  y  thick,  5  feefc  wide.    During  the  con- 
.struction  the  dam  wras  planked  first  about 
30  feet  on  the  incline  ;  a  space  was  then 
left  of  about  16  feet  width  by  sufficient 
length,  through  which  the  water  flowed ; 
and  the   balance   of  the   dam  was   then 
completed.     A  plank-flap  was  then  made 
for  the  opening,  and  when  every  thing 
was  ready,  it  was  shut  down,  and  the 
pond  filled.     The  dam  was  built  under 
the   direction   of   the   late   Mr.    John 
Chase,  and  since  its  construction  the 
greatest  depth  of  water  passing  over 
the  crest  during  a  freshet  was  12'  6". 

Some  years  after  the  construc- 
tion of  this  dam  it  was  found  that 
the   overfall  of   water  from   its 
crest    was    wearing    away    the 
ledge  and  jeopardized  the  foun- 
dation of  the  dam.    An  apron 
(Fig.    835)    was     therefore 
constructed   of   crib-work, 
sheathed  with  plank,  add- 
ing stability  to  the  struct- 
ure,    and     discharging 
the  water  more  nearly 
in  the  line  of  the  river 
current. 

Fig.    836   is    a 
section  of  part  of 
the  dam  across  the 
Merrimack  Eiver,  at 
Lowell,    built    under 


ENGINEERING  DRAWING. 


377 


FIG.  836. 


378 


ENGINEERING  DRAWING. 


SCALE  : 
A  inch  =  1  foot. 


.  837. 


the  direction  of  Mr.  James  B.  Francis.     It  was  laid  dry,  with  the  exception 
of  the  upper  face  and  coping,  which  was  laid  full  in  cement. 

The  horizontal  joints  at  the  crest  were  run  in  with  sulphur.  The  coping- 
stones  were  doweled  to  the  face  and  together,  and  clamped  to  an  inclined  stone 
on  the  lower  slope  ;  the  end-joint  between  these  stones  was  broken  by  making 
every  alternate  lower  stone  longer,  and  the  upper  shorter,  than  shown  in  the 
drawings. 

The  Cohoes  dam  (Fig.  837)  was  built  under  my  direction,  directly  below  an 
old  dam  of  somewhat  similar  construction  to  that  of  Holyoke.  The  old  dam 
had  become  very  leaky  and  worn,  and  the  overfall  had  in  many  places  cut  deep 

into  the  rock,  and  in  some 
places  within  the  line  of  the 
dam.  It  was  therefore  pro- 
posed to  make  the  new  dam, 
as  a  roll  to  the  old  one,  to 
discharge  the  water  as  far 
from  the  foot  of  the  dam  as 
possible,  and  to  keep  the  old 
dam  for  the  protection  of  the 
new.  The  exterior  of  the 
dam  was  of  rock-faced  ash- 
lar ;  the  caps  were  in  single 
lengths  of  10  feet,  and  none 
less  than  15"  thick  and  2  feet  wide  ;  they  were  doweled  together  with  two 
galvanized  wrought-iron  dowels  each.  The  whole  work  was  laid  full  in  cement, 
the  20"  wall  next  the  old  dam  being  laid  distinct  without  bond  into  the  rest 
of  the  work.  The  whole  was  brought  up  to  the  outline,  to  receive  the  cap- 
stones, which  were  bedded  in  cement  ;  the  top-joints  were  then  run  or  grouted 
in  neat  cement,  to  within  about  6"  of  the  top  of  the  stone,  which  was  after- 
ward run  in  with  sulphur.  Entire  length  of  overfall,  1,443  feet ;  average 
depth  below  crest  of  dam,  12  feet. 

Where  the  body  of  water  which  may  at  any  time  discharge  over  the  dam  is 
large  and  the  fall  high,  it  is  especially  desirable  to  secure  a  location  where  the 
overfall  can  be  upon  solid  rock.  If  there  be  ledge  at  the  side  of  the  river,  and 
none  can  be  found  in  the  channel,  it  is  often  better  to  make  a  solid  dike  across 
the  river  and  above  the  level  of  freshets,  and  cut  the  overfall  out  of  the  bank. 
When  from  any  circumstances  the  dam  can  have  only  an  earth  foundation,  an 
artificial  apron,  or  platform  of  timber  or  rock,  is  to  be  made,  on  which  the 
water  may  fall,  or  the  high  fall  may  be  broken  up  by  a  succession  of  steps.  In 
some  cases,  a  roll  or  incline,  like  that  given  in  Croton  dam,  is  extended  to  the 
bed  of  the  stream,  and  continued  by  an  apron.  The  water  thus  rolls  or  slides 
down,  and  takes  a  direction,  as  it  leaves  the  apron,  parallel  with  that  of  the 
bed  of  the  streanio  But  care  must  be  taken  to  protect  the  outer  extremity  of 
the  apron  by  sheet-piling  and  heavy  paving,  as  the  current,  by  its  velocity, 
takes  with  it  gravel  and  all  small  rocks,  and  undermines  the  apron. 

Dams  or  dikes  are  often  made  entirely  of  compacted  earth  ;  sometimes  with 
a  puddle-wall  of  clay  in  the  center,  as  in  the  reservoir  embankment  (Fig.  860), 


ENGINEERING  DRAWING. 


379 


or  a  sheet-piling.  Dikes  across  salt 
marshes  are  made  of  material  taken 
from  the  marsh  at  some  distance 
from  the  site  of  the  dike,  well  packed 
in  thin  layers  on  a  base  prepared  on 
the  soil  without  excavation.  Sand 
and  gravel,  being  heavier  than  the 
moist  material,  break  through  it  and 
settle  to  the  bottom,  involving  often 
the  construction  of  a  large  embank- 
ment, while,  by  the  use  of  a  homo- 
geneous material,  the  foundation  is 
not  displaced  but  compressed. 

Fig.  838  is  a  section  of  the  dike 
or  embankment  for  the  Ashti  Tank 
or  Reservoir,  constructed  for  retain- 
ing water  for  irrigation  purposes  in 
India.  The  following  is  an  abstract 
of  the  description  of  the  work  given 
in  the  "Minutes  of  the  Proceedings 
of  the  Institute  of  Civil  Engineers," 
vol.  Ixxvi : 

u  The  net  supply  available  for  irriga- 
tion may  be  calculated  thus  : 

Available  capacity 

of  tank 1,348,192,450  cub.  ft. 

Deduct  loss  by  evap- 
oration, etc 233,220,240  " 

Net  supply  available 

for  irrigation. ..   1,114,972,210      " 

"  Area  of  catchment  basin  nearly  92 
square  miles." 

The  total  length  of  the  dam  is 
12,709  feet ;  the  breadth  at  the  top, 
which  is  uniform  throughout,  six 
feet  ;  breadth  at  full  supply-level, 
42  feet ;  height  of  the  top  of  the 
dam  above  full  supply-level,  12  feet ; 
greatest  height  of  dam,  58  feet. 
The  seat  of  the  dam  throughout  was 
cleared  of  vegetable  mold,  stones, 
and  loose  material,  all  trees  and 
shrubs  with  their  roots  being  com- 
pletely grubbed  or  dug  out.  The 
puddle-trench  laid  in  the  natural 


JUL 


380  ENGINEERING  DRAWING. 

ground  is  rectangular  in  cross-section,  10  feet  in  width,  excavated  through 
various  materials  to  a  compact  water-tight  bed,  and  then  filled  in  with  puddle 
material,  consisting  of  two  parts  of  muram  or  sand,  and  three  parts  of  black 
soil,  carefully  mixed  and  worked  by  treading  with  the  feet,  and  then  kneaded 
into  balls  and  thrown  or  dashed  into  the  trench  in  layers  up  to  12  inches  in 
thickness.  The  puddle  was  brought  to  a  level  of  one  foot  above  the  ground. 
Across  the  river  the  trench  was  cut  down  to  the  rock  and  filled  with  concrete. 
The  general  distribution  of  the  material  of  the  dam  is  shown  in  the  figure. 
The  central  core  is  formed  of  the  best  black  soil  attainable ;  on  each  side,  ex- 
tending to  the  surface  of  the  mixed  material,  brown,  reddish,  or  white  earth  is 
used.  The  outer  part  of  the  dam  is  formed  of  a  mixture  of  equal  parts  of  black 
soil  and  muram,  but  where  muram  was  difficult  to  obtain,  and  sand  plentiful, 
the  latter  was  substituted  for  muram  in  the  mixture.  The  black  soil  may  be 
described  as  a  clayey  earth,  tenacious  and  adhesive  when  wet — a  product  of 
the  decomposition  of  volcanic  rock.  The  brown  and  reddish  soils  are  of  a 
clayey  nature,  but  contain  admixtures  of  fine  sand,  kunkun  nodules,  and  thin 
layers  of  fine  grains  of  lime.  The  white  soil  consists  of  finely  powdered  parti- 
cles of  a  grayish  color,  similar  to  wood-ashes,  which  when  dry  possesses  little 
adhesion,  but  when  wet  is  adhesive. 

The  various  soils  were  laid  in  the  work  in  layers  eight  inches  in  thickness, 
every  layer  being  thoroughly  watered  and  rolled  with  iron  rollers.  The  outer 

slope  was  protected  by  a  mixture  of 
equal  parts  of  soil  and  reddish  mu- 
ram, and  with  sods  of  grass,  laid 
about  three  feet  apart,  which  in  time 
extended  over  the  whole  slope. 

The  inner  slope  is  protected  from 
the  action  of  the  waves  by  being 
pitched  or  faced  with  dry  stone,  set 
by  hand,  and  laid  on  a  layer  of 
coarse  muram.  The  stones  of  the 
FlQ-  839-  pitching  were  bedded  on  the  slope, 

and  were   laid  with  their  broadest 

end  downward  (Fig.  839),  each  stone  being  roughly  squared  with  the  hammer, 
and  touching  for  at  least  three  or  four  inches.  The  interstices  were  then 
packed  with  small  stone-chippings,  and  finished  off  with  muranio 

Head-gates  are  constructions  necessary  to  control  the  flow  from  the  river- 
pond  or  reservoir  into  the  canal  or  conduit  by  which  the  water  is  to  be  con- 
veyed and  distributed  for  the  purposes  to  which  it  is  to  be  applied.  The  top 
of  the  works  should  therefore  be  entirely  above  the  level  of  the  highest  freshets, 
that  no  water  may  pass,  except  through  the  gates ;  and  it  is  better  that  the 
opening  of  the  gates  should  be  entirely  below  the  level  of  the  top  of  the  dam, 
to  prevent  as  much  as  possible  the  passage  of  drift  and  ice,  which  are  often  ex- 
cluded by  booms  and  racks  placed  outside  the  gates. 

Figs.  840  and  841  are  drawings,  in  plan  and  detail,  of  the  head-gates,  and 
the  machinery  for  hoisting  them,  at  the  Cohoes  Company's  dam. 

It  will  be  seen,  by  reference  to  the  plan,  that  there  are  ten  gates.     The 


ENGINEERING  DRA1 


382 


ENGINEERING   DRAWING. 


ENGINEERING   DRAWING. 


383 


dimensions  of  four  are  8'  x  6'  6" ;  and  six,  8'  x  9',  in  the  clear — all  of  which 
can  be  hoisted  by  machinery  connected  with  a  turbine-wheel  at  a,  or  separately 
by  hand.  At  b  there  is  an  overfall,  at  the  same  height  as  the  dam,  over  which 
any  drift  that  is  brought  against  the  gate-house  is  carried.  At  c  there  is  a 
similar  overfall  within  the  gates,  and  another  at  d,  by  which  any  sudden  rise 
of  the  level  of  the  canal  is  prevented.  At  e  there  is  a  gate  for  drawing  down 
the  pond,  and  another  at  /,  for  drawing  off  by  the  canal,  both  raised  and  low- 
ered like  the  head-gates. 

The  head-gates  are  of  solid  timber  bolted  together,  moving  in  cast-iron 
guides  set  in  grooves  in  the  stone  ;  in  front  of  these  grooves  there  is  another 
set  of  grooves  (gg}>  which  are  intended  for  slip-planks  or  gates,  to  be  put  in 
whenever  it  is  necessary  to  shut  off  the  water  from  the  gates  themselves  in  case 
of  repairs. 

Hoisting  Apparatus. — To  each  gate  there  are  strongly  bolted  two  cast-iron 
racks,  geared  into  two  pinions  on  a  shaft  extending  across  the  gate-space,  and 


FIG.  842. 


FIG.  843. 


supported  on  cast-iron  standards  on  the  piers.     At  one  extremity  of  this  shaft, 
there  is  a  worm-wheel,  driven  by  a  worm  or  screw  on  a  shaft  perpendicular  to 


384  ENGINEERING  DRAWING. 

the  pinion-shaft.  The  worm-shaft  can  be  driven  either  by  a  hand-wheel  at  one 
end,  or  by  the  friction -bevel  at  the  other.  The  friction-bevel  can  be  driven  in 
either  direction  by  being  brought  in  contact  with  one  or  other  of  the  friction- 
bevels  on  a  shaft  extending  the  whole  length  of  the  gate-house,  and  in  gear 
directly  with  the  small  turbine  at  a.  The  small  turbine  draws  its  supply 
through  a  pipe,  built  in  the  walls,  and  opening  into  the  space  between  the 
gates  and  the  slip-plank  groove. 

Figs.  842  and  843  are  the  front  elevation  and  section  of  the  gates  of  Farm 
Pond,  Sudbury  River  Conduit,  Boston  Water- Works.  The  main  web  or  plate 
of  the  gate  is  !£*  thick,  the  ribs  6"  deep,  the  gate-stems  %\"  diameter.  The 
nuts  by  which  the  gates  are  raised  are  geared  together,  and  actuated  by  a 
double  crank.  For  smaller  gates  it  is  usual  to  have  but  a  single  stem,  and  the 
nut  in  a  hand-wheel  on  top  of  the  standard.  The  gates  and  guides  are  faced 
with  brass,  about  Ty  thick. 

Gates  of  this  form  are  very  common,  consisting  of  plates  of  cast-iron 
strengthened  by  ribs  ;  the  guides  are  also  of  cast-iron,  bolted  to  the  masonry. 
The  faces  of  the  gates  and  guides  are  usually  covered  by  brass  plates,  as  iron 
faces  become  rusty.  When  the  gates  are  small,  there  is  usually  but  one  stem. 
Often,  instead  of  nuts  and  screws,  racks  and  pinions  are  used  ;  and  with  heavy 
wooden  gates,  requiring  but  little  use,  the  gates  are  raised  by  chains  over  a  bar- 
rel, by  hand-spikes,  and  ratchets  to  hold  the  gates  in  position  as  they  are  raised. 

Canals. — The  sections  of  canals  depend  upon  the  purposes  to  which  they 
are  to  be  applied,  whether  for  navigation  or  for  power  ;  if  for  navigation, 
reference  must  be  had  to  the  class  of  boats  for  which  they  are  intended  ;  if  for 
power,  to  the  quantity  of  water  to  be  supplied,  and  sundry  precautions  of  con- 
struction. 

Fig.  844  is  a  section  of  the  Erie  Canal :  width  at  water-line,  70  feet ;  at 
bottom,  28  feet ;  depth  of  water,  7  feet ;  width  of  tow-path,  14  feet.  It  will 
be  observed  that  the  slopes  are  graveled  and  paved,  and  that  the  edge  of  the 


FIG.  844. 

tow-path  is  paved  with  cobble-paving,  and  the  path  graveled.  The  smaller 
canals  of  this  State  and  of  Pennsylvania  are  generally  40  feet  wide  at  water- 
line,  and  4  feet  deep ;  the  Delaware  and  Raritan,  75;x  7';  the  Chesapeake  and 
Delaware,  66'  x  10';  the  ship-canals  of  Canada,  10  feet  deep  and  from  70  to 
190  feet  wide. 

The  dimensions  for  canals  for  the  supply  of  mills  depend — first,  on  the 
quantity  of  water  to  be  delivered.  Their  area  of  cross-section  should  be  such 
that  the  average  velocity  of  flow  should  not  exceed  two  feet  per  second,  and  in 
northern  climates  less  velocity  than  this  would  be  still  better  ;  it  should  always 
be  such  that  during  the  winter  the  canals  may  be  frozen  over,  and  remain  so, 
to  prevent  the  obstruction  from  drift  and  anchor-ice  in  the  water-wheels.  The 


ENGINEERING  DRAWING. 


385 


usual  depths  of  the  larger  canals  are  from  10  to  15  feet ;  with  such  depths  the 
cover  of  ice  which  reduces  the  section  by  the  amount  of  its  thickness  does  not 
materially  increase  the  velocity  of  flow,  nor  diminish,  consequently,  very  per- 
ceptibly the  available  head. 

Fig.  845  is  a  section  of  the  Northern  Canal,  at  Lowell,  Mass.,  which  may 
be  considered  a  model  for  large  works.     The  width  at  water-line  is  103  feet, 


FIG.  845. 

and  the  depth  16',  and  is  intended  for  an  average  flow  of  2,700  cubic  feet  per 
second.  The  fall  in  the  whole  length  of  4,300  feet  is  between  2"  and  3";  when 
covered  by  ice,  about  4".  The  sides  are  walled  in  dry  rubble,  and  coped  by 
split  granite.  It  will  be  observed  that  the  portion  above,  and  about  three  feet 
below,  the  water-line,  or  between  the  limits  of  extreme  fluctuations  of  level, 
is  laid  plumb,  that  the  ice  may  have  as  free  a  movement  as  possible  vertically. 

Fig.  846  is  a  sec- 
tion, on  a  scale  of  -J" 
=  1  foot,  of  the  river- 
wall  of  this  same  ca- 
nal, where  the  canal 
passes  out  into  and 
occupies  a  portion  of 
the  river-channel,  and 
the  depth  of  water  in 
the  canal  is  greater 
than  in  the  above  sec- 
tion. The  main  wall 
is  in  dry  masonry, 
faced  on  river -side 
with  rough-faced  ash- 
lar, pointed  beds  and 
end-joints.  The  in- 
side lining  is  of  two 
courses  of  cement- 
wall,  the  dry  rubble 
backing  being  first 
laid,  then  pointed  FIG.  846. 

with  cement,  against 

which  is  laid  the  first  cement  lining,  which  is  plastered  on  the  inside,  and  the 
interior  wall  is  then  laid ;  the  granite  inside  wall,  above  lining,  is  also  laid  in 
cement. 

25 


386 


ENGINEERING  DRAWING. 


FIG.  847. 


SCALE  :  A"  =  1  foot. 


FIG.  848. 

Locks  of  Canals. — Figs.  847  and  848  are  portions  of  plan  and  vertical  sec- 
tion of  locks,  taken  from  the  general  plans  for  timber  locks  on  the  Chemung 

Canal.      They  represent  the  half  of 
upper  gates. 

Fig.  849  is  a  section ,  of  one  side 
of  the  lock  of  the  same. 

Fig.  850  is  the  plan  of  a  portion 
of  one  of  the  enlarged  locks  of  the 
Erie  Canal,  showing  one  of  the  upper 
gates  and  the  side-walls. 

Fig.  851  is  a  cross-section  of  one 
of  the  same  locks,  showing  the  cul- 
vert in  the  center  between  the  locks, 
FIG.  849.  used  for  the  supply  of  the  waste  of 


ENGINEERING  DRAWING. 


the  lower  level,  to  preserve  the  proper  height  of 
by  gates  in  the  upper  level. 


' 

\\\ 

E3   v—  \ 

IA  v 

\ 

tf  this  level  controlled 


FIG.  850. 


I L 


J L 


FIG.  851. 


SCALE  :  -fs"  =  1  foot. 


FIG.  852. 


full  size. 
FIG.  853. 


Fig.  852  is  a  drawing,  in  outline,  of  the  hollow  quoin  of  the  lock-gate,  on  a 
scale  of  -^T  full  size  (Chemung  Canal). 

Fig.  853  is  a  plan  and  elevation  of  pintal  for  heel-post  of  lock,  with  a  sec- 


388  ENGINEERING  DRAWING. 

tion  of  the  bottom  of  the  post.  The  pintal  is  imbedded  in  bottom  timber  or 
stone,  as  the  case  may  be. 

Fig.  854  is  a  plan  and  elevation  of  the  strap  for  the  upper  part  of  heel-post. 

Extracts  from  lock  specifications  ("  New  York  State  Canals,"  1854)  : 

"  Locks  to  be  composed  of  hydraulic 
stone  masonry,  placed  on  a  foundation 
of  timber  and  plank.  The  chamber  to 
be  18'  wide  at  the  surface  of  the  water 
in  the  lower  level,  and  110'  long  be- 
tween the  upper  and  lower  gate-quoins. 
The  side- walls  to  extend  21'  above  the 
upper  gate-quoins,  and  14'  below  lower 
gate-quoins.  If  the  bottom  is  of  earth, 
and  not  sufficient  to  support  the  foun- 
dation, then  bearing-piles  of  hard  wood,, 
not  less  than  10"  diameter  at  small  end, 
shall  be  driven,  to  support  the  founda- 

-  854.  tion.     There  shall  be  four  rows  of  piles- 

under  each  main  wall,  and  one  row  in 

center  of  lock  ;  the  piles  shall  be  driven  in  rows,  at  3'  from  center  to  center.  The  piles  to 
support  the  wing  and  breast-walls  and  wing  buttresses,  and  also  under  the  miter-sills,  to  be 
driven  in  rows  to  conform  to  the  form  and  shape  of  the  same.  The  heads  of  the  piles  to 
be  cut  off  smooth  and  level,  to  receive  the  foundation  timbers.  The  foundation  timbers 
to  be  12''  x  12",  and  of  such  lengths  as  will  extend  from  and  cover  the  outside  piles,  and 
to  be  treenailed  with  a  2"  white-oak  or  white-elm  treenail,  24"  long,  to  each  pile. 

"  If  the  bottom  is  of  earth  sufficiently  compact  and  firm  to  support  the  foundation 
without  bearing-piles,  then  the  foundation  shall  be  composed  of  timber,  12"  thick  and  not 
less  than  10"  wide,  counterhewed  on  upper  side,  timbers  to  average  12"  wide,  to  be  placed 
at  uniform  distance,  according  to  their  width,  so  as  to  occupy  or  cover  at  least  £  of  the  area 
of  the  foundation,  and  under  the  lower  miter-sill  to  be  placed  side  by  side :  in  all  cases  to 
be  of  sufficient  length  to  extend  across  the  lock  to  the  back  line  of  the  center  buttresses, 
and  at  the  head  and  foot  to  the  rear  or  back  line  of  wing-walls.  The  timber  under  the 
lower  miter-sill  to  be  of  white  oak,  white  elm,  or  red  beach,  the  other  foundation  and 
apron  timber  to  be  of  hemlock.  The  foundation  to  be  extended  3'  above  the  face  of  the 
main  wall  at  the  head  of  the  lock,  and  at  the  foot  from  25'  to  30'  below  the  exterior  wing — 
that  portion  of  the  spaces  between  the  timbers  in  all  cases  to  be  filled  with  clean  coarse 
gravel,  well  rammed  in,  or  concrete.  In  cases  where  rock  composes  the  bottom  of  the 
lock,  the  foundation  timbers,  if  required,  shall  be  10"  thick  under  the  lower  miter-sill,  and 
8"  thick  at  other  places.  Where  the  rock  is  of  such  a  character  that  timber  is  not  required 
for  the  foundation,  the  same  shall  be  excavated  smooth  and  level,  and  the  first  course  of 
stone  well  fitted  to  the  rock. 

"  Sheet- Piling. — In  all  cases  where  rock  does  not  occur,  there  shall  be  a  course  at  the 
head  of  the  foundation,  under  each  miter-sill,  and  at  the  lower  end  of  the  wings,  and  at  the 
lower  end  of  the  apron,  to  be  from  4'  to  6'  deep  as  may  be  required — in  each  to  extend 
across  the  whole  foundation.  The  sheet-piling  to  be  of  2"  hemlock  plank,  lined  with  1" 
pine  boards.  Ditches  are  to  be  excavated  to  receive  the  sheet-piling,  which  are  to  be 
placed  edge  to  edge,  and  the  top  well  secured  to  the  foundation  timber ;  the  spaces  to  be 
filled  up  with  fine  hard  gravel,  well  puddled  in,  or  with  concrete. 

"  Flooring. — A  course  of  2£"  pine  or  hemlock  plank  to  be  laid  over  the  whole  of  the 
foundation  timbers,  except  a  space,  3'  wide,  under  the  lace-line  of  each  wall  to  be  2£" 
white  oak :  the  whole  to  be  well  jointed,  and  every  plank  to  be  treenailed  with  two  white- 


ENGINEERING  DRAWING.  389 

oak  treenails  at  each  end,  and  at  every  3'  in  length,  to  enter  the  timber  at  least  5",  or  with 
wrought-iron  spikes,  treenails  to  fill  1J-"  bore.  Platform  for  the  upper  miter-sill  to  be  5' 
10"  wide,  and  6'  high  above  foundation,  and  to  extend  across  from  side-wall  to  side-wall, 
to  be  composed  of  masonry,  coped  with  white- oak  timbers,  which  are  to  extend  6"  into 
each  side-wall.  The  timbers  to  be  12"  deep  and  14"  wide,  covered  with  two  courses  of 
\\"  white-oak  plank.  Miter-  sills  to  be  of  best  white-oak  timber,  9"  thick,  to  be  well  jointed, 
and  bolted  to  the  foundation  or  platform  timbers,  as  the  case  may  be,  with  bolts  of  iron, 
20"  long,  1"  x  1",  well  ragged  and  headed,  eight  bolts  to  each  side. 

"  Masonry. — The  main  walls,  for  21'  6"  in  length,  from  wing-buttresses  at  the  head, 
and  32'  at  lower  end,  to  be  9'  8£"  thick,  including  recesses,  and  for  the  intermediate  space, 
V  8£"  thick,  with  three  buttresses  projecting  back  2V,  and  9'  long  at  equal  distances  apart. 
The  quoin-stones,  in  which  the  heel-post  is  to  tarn,  shall  not  be  less  than  4'  6"  in  length  in 
line  of  the  chamber,  to  be  alternately  header  and  stretcher.  The  recesses  for  the  gates  to 
be  20"  wide  at  top  of  wall,  12'  long,  with  sub-recesses,  9"  wide,  6'  high,  10'  long,  for  the 
valve-gates.  Breast-wall  to  commence  5'  below  upper  end  of  foundation,  5'  wide,  8' high, 
finished  with  a  coping  of  cut  stone.  The  interior  wing-walls,  and  exterior  wing  from  main 
walls  to  the  termination  of  first  curve,  to  be  7'  6"  thick,  and  the  running  curve  of  exterior 
wing  to  be  6'  thick  on  the  foundation. 

"  Culvert  between  Locks. — In  such  cases  as  may  be  required,  a  culvert  shall  be  con- 
structed, to  pass  the  water  from  the  upper  to  the  lower  level,  as  follows:  A  foundation 
of  suitable  timber  and  plank,  as  for  lock-walls,  and  covering  all  the  space  between  the 
lock-foundations,  shall  be  put  down.  Three  apertures  for  the  sluice-way  shall  be  made 
in  the  head-wall  with  cut-stone  jambs,  grooves  to  be  cut  in  the  jambs  for  the  sluice- 
gates, and  the  coping  to  form  a  recess,  corresponding  with  the  grooves  in  the  jambs ; 
grooves  to  be  cut  on  the  top  and  bottom  coping,  1"  deep,  to  secure  the  jambs.  The 
bottom  of  the  aperture  to  be  of  cut  stone,  with  lower  corner  beveled  off,  over  which  the 
water  will  fall  into  the  well,  the  bottom  of  which  shall  be  covered  with  a  sheeting  of 
cut  stone,  6"  thick.  The  apertures  to  be  3'  6"  deep,  placed  immediately  below  the  coping- 
stone,  and  4'  long.  Suitable  gates  of  plank,  for  regulating  the  water  in  passing  the  sluice, 
to  be  prepared  ;  the  well  to  commence  on  the  foundation,  to  be  made  of  substantial  hy- 
draulic masonry. 

"  Second  flooring  of  seasoned  2"  first-quality  white-pine  plank,  to  be  well  jointed,  and 
laid  on  the  foundation  between  the  walls,  from  the  breast-wall  to  lower  end  of  main  wall, 
and  also  on  the  floor  of  the  well,  to  be  close  and  firmly  jointed  to  miter-sills  and  walls,  so 
as  to  make  a  water-tight  flooring.  The  plank  to  butt,  or  the  end-joints  to  come  to  the 
center  of  a  foundation  timber,  and  each  plank  to  be  treenailed  with  two  treenails  at  end 
and  two  at  every  3'  intermediate :  treenails  10"  long,  to  fill  1J"  bore. 

"  Gates. — The  framing  to  be  made  of  best  quality  white-oak  timber;  the  cross-bar  to 
be  framed  into  heel  and  toe  posts  with  double  tenons,  each  tenon  to  be  7"  long,  and  thick- 
ness equal  to  the  thickness  of  the  bar,  and  secured  with  wrought-iron  Ts,  well  bolted. 
The  heel  and  the  posts  to  be  framed  to  the  balance-beam  by  double  tenons,  and  secured  by  a 
wrought-iron  strap  and  balance-rod,  from  the  top  of  the  beam  to  the  under  side  of  the  upper 
bar.  The  lower  ends  of  the  heel-posts  to  be  banded  with  wrought-iron  bars;  the  collar 
and  other  hangings  to  be  of  wrought-iron,  secured  together  with  a  double  nut  and  screw, 
and  to  the  coping  by  bedding  the  depth  of  the  iron  in,  and  by  screw-bolts  fastened  with 
sulphur  and  sand-cement.  The  pivots  and  sockets  which  support  the  heel-posts  to  be  of 
best  cast-iron;  a  chilled  cast-iron  elliptical  ball,  2V  horizontal,  and  1"  vertical  diameter, 
to  be  placed  on  the  pivot  and  in  the  socket  of  each  heel-post,  to  facilitate  the  movement 
of  the  gate.  The  gates  to  be  planked  with  seasoned  first-quality  2"  white-pine  plank, 
jointed,  grooved,  and  tongued— tongues  of  white  oak — the  plank  to  be  secured  by  6" 
pressed  spike.  On  the  chamber-side  of  the  gates,  fenders  of  white-oak  plank,  to  be  put  on 
with  pressed  spike." 


390  ENGINEERING  DRAWING. 

Water,  ponded  by  dams,  and  conveyed  by  canals  for  use  as  mill-power,  is 
carried  within  the  workshops  or  manufactories,  to  be  applied  on  water-wheels, 
by  some  covered  channels.  These  channels,  although  of  various  forms,  are 
usually  designated  as  flumes.  The  common  form  of  a  flume  for  the  convey- 
ance of  water  to  breast,  overshot,  or  undershot  wheels,  is  of  a  rectangular  sec- 
tion, framed  with  sills,  side-posts,  and  cap,  and,  if  large  section  is  required, 
intermediate  posts  are  set  in.  The  sills  are  set,  and  earth  well  rammed  in  the 
spaces  between  them  ;  the  bottom  plank  is  then  laid,  posts  and  cap  framed  with 
tenon  and  mortice,  set  and  pinned,  and  the  plank  is  then  firmly  spiked  on  the 
outside  of  posts  and  caps.  The  planks  are  usually  nearly  green,  jointed,  and 
brought  to  close  joints  ;  the  size  of  timbers  will  depend  on  the  depth  beneath  the 
soil,  or  the  insistent  load.  Within  the  mill,  and  just  above  the  wheel,  the 
flume  is  framed  without  a  cover,  and  the  posts  and  side-planks  are  brought  above 
the  level  of  the  water.  This  open  flume  is  termed  the  penstock,  especially  neces- 
sary, in  the  class  of  wheel  above  referred  to,  to  secure  the  full  head  of  water. 

Many  flumes  are  made  of  a  circular  section,  pipes  of  iron,  or  wood.  For 
the  conveyance  of  water  to  turbine-wheels,  wrought-iron  pipes  are  almost  inva- 
riably used.  Cast-iron  is  also  sometimes  used,  with  flange,  or  hub  and  spigot- 
joints.  Plank-pipes  are  also  used,  made  with  continuous  staves,  and  hooped 
with  wrought-iron  ;  these  constructions  are  much  cheaper,  and  serve  a  very 
good  purpose.  The  head-gates  of  flumes  are  placed  at  the  head  of  the  flumes, 
in  a  recess  back  from  the  face  of  the  canal,  with  racks  in  front  to  prevent  the 
passage  of  any  drift  that  might  obstruct  or  injure  the  wheel.  The  total  area 
of  passages  through  the  racks  should  liberally  exceed  the  area  of  cross-section 
of  the  flume,  not  only  on  account  of  the  extra  lateral  friction  of  the  rack-bars, 
but  also  on  account  of  their  liability  to  become  obstructed.  Sometimes  two 
sets  of  racks  are  placed  in  front  of  the  flumes,  especially  for  turbines  and  react- 
ing wheels  :  a  coarse  rack  with  wide  passages,  say  2"  spaces  outside,  and  a  finer 
one  inside,  say  of  f"  to  f "  spaces.  The  head -gates  to  the  flume,  directly  back 
of  the  racks,  in  their  function  are  like  the  head-gates  at  the  dam,  and  are  simi- 
lar in  construction — strong  plank  gates,  moving  in  slides,  vertically  or  horizon- 
tally, with  a  paddle-gate  in  them,  to  fill  the  flume  when  empty,  so  that  the 
gates  themselves  may  be  opened  without  any  pressure  due  to  a  difference  of 
head  outside  and  inside  of  the  gates,  and  also  to  prevent  any  damage  to  the 
flume  by  the  water-ram,  which  might  result  from  a  too  sudden  filling  of  the 
flume  by  the  opening  of  a  large  gate  suddenly. 

Fig.  855  is  the  elevation  and  section  of  the  head-gates  manufactured  at 
Holyoke,  Massachusetts.  G  G  are  plank  gates,  sliding  laterally,  moved  by 
two  pinions,  working  into  racks  on  top  and  bottom  of  gates,  turned  by  a  hand- 
spike. P  is  the  paddle-gate  ;  R,  the  rack  ;  F,  the  flume,  or  plank-pipe  ;  A, 
air-pipe,  for  the  escape  of  air  from  the  flume  while  being  filled. 

Conduits  for  the  supply  of  water  to  cities  and  towns  are  of  masonry,  or  cast 
or  wrought  iron  pipes.  Their  capacity  to  deliver  the  required  quantity  depends 
upon  the  area  and  form  of  cross  section,  and  the  velocity  of  flow  due  to  the 
loss  of  head  or  of  pressure  permissible  ;  this  velocity  being  due  primarily  to 
gravity,  but  largely  modified  by  conditions  of  structure,  as  the  kind  and 
amount  of  wetted  surface,  and  length  and  directness  of  line. 


ENGINEERING  DRAWING. 


391 


392 


ENGINEERING  DRAWING. 


Fig.  856  is  a  cross-section  of  the  main  conduit  of  the  Nassau  Water- Works 
for  the  supply  of  the  city  of  Brooklyn,  Long  Island.  The  width  is  10"  at  the 
springing  of  the  arch  ;  the  side-walls  3  feet  in  height ;  versed  sine  of  invert, 
8*  ;  height  of  conduit  in  center,  8'  8"  ;  fall  or  inclination  of  bottom,  1  in 
10,000. 

In  preparation  of  the  foundations  the  contract  specifications  required  a  bed  of  concrete 
to  be  first  laid,  15'  wide;  but,  when  the  water  was  very  troublesome,  it  was  found  neces- 
sary to  lay  a  platform  of  plank  for  the  concrete.  The  side-walls  are  of  stone,  except  aD 
interior  lining  of  4"  brickwork.  The  arch  is  brick,  12",  and  the  invert  4"  thick.  The 
outside  of  arch,  as  it  was  finished,  and  each  wall,  were  plastered  over  on  the  outside  with 
a  thick  coat  of  cement-mortar.  The  concrete  was  formed  from  clean  broken  stone,  broken 
so  as  to  pass  through  a  2"  ring;  2  to  2£  measures  of  broken  stones  were  mixed  with  1 
measure  cement-mortar.  The  centers  of  the  arching  were  not  allowed  to  be  struck  until 
the  earth  had  been  well  packed  in  behind  the  side-walls  and  half-way  up  the  arch.  In 
both  cuttings  and  embankments  the  arch  was  covered  with  4  feet  of  earth,  with  a  width 
of  8  feet  at  top,  and  slopes  on  each  side  of  1-J-  to  1,  covered  with  soil  and  seeded  with  grass. 


Fio.  856. 


FIG.  857. 


Fig.  857  represents  a  section  of  the  Oroton  Aqueduct,  in  an  open  rock-cut. 
The  width  at  spring  of  arch,  7';  versed  sine  of  invert,  6*;  height  of  conduit, 
8'  6";  fall  or  inclination  of  bottom,  about  1  in  5,000. 

The  bottom  is  raised  with  concrete  to  the  proper  height  and  form  for  the  inverted 
arch,  of  a  single  course  of  brick ;  the  side-walls  are  of  stone,  laid  in  cement,  plastered, 
and  faced  with  a  single  course  of  brick;  the  arch  is  semicircular,  of  brick  two  courses 
thick,  with  spandrel  backing  nearly  to  the  level  of  the  crown,  and  earth  filled  on  the  top. 
In  earth-cuts  or  embankments,  side-walls  were  constructed  of  stone,  in  cement;  and  in 
embankments  the  whole  structure  rested  on  dry  rubble-walls,  built  up  from  solid  earth- 
foundations. 

At  the  crossing  of  the  Harlem  Eiver,  as  the  bridge  was  depressed  below 
the  level  of  the  aqueduct,  the  water  was  conveyed  by  two  cast-iron  pipes, 
a  a,  3'  in  diameter,  Fig.  858  ;  but,  as  the  demand  for  water  increased  in  the 
city,  the  obstruction  caused  by  lack  of  capacity  in  these  pipes  has  made  nec- 
essary the  introduction  of  a  larger  pipe,  which  has  been  made  of  wrought- 
iron,  -J"  thick  and  7'  6£"  in  diameter ;  this  is  supported  by  cast-iron  columns 


ENGINEERING  DRAWING. 


393 


which  admit  of  a 
rocking  movement, 
and  slip- joints  are  also 
made  in  the  pipe  to 
compensate  for  any 
expansion  or  contrac- 
tion by  changes  of 
temperature.  The 
pipes  are  inclosed  in 
a  long  chamber  or 
passage,  extending 
the  whole  length  of 
the  bridge,  covered 
by  a  brick  arch,  laid 
in  cement  with  a 
cover  of  asphalt,  and 
a  brick  pavement 
over  all,  affording  a 
wide  promenade  pro- 
tected on  each  side 
by  cast-iron  railings, 

fastened  to  the  coping-stones,  CO.    A  A  are  the  arch-stones  of  the  bridge. 
Fig.   859  is  a  section  of  the  conduit  of  the  Boston  Water- Works.      The 

inside  section  is  equal  to  a  circle  8£  feet  diameter,  and  is  uniform  throughout 


FIG.  858. 


FIG.  859. 


except  in  tunnels.      The  exterior  lines  vary  according  to   the  material  on 
which  it  is  built  and  the  cover  or  load  on  the  top.     The  section  given  may  be 


394 


ENGINEERING  DRAWING. 


considered  the  general  one,  resting  on  a  bed  of  concrete,  with  masonry  sides  ; 
brick  lining  at  sides  and  invert  at  bottom,  with  an  8"  arch  at  top  for  a  4' 
cover,  and  12"  for  exceptional  depths  or  under  railway-tracks.  The  lower 
corners  were  of  brick,  of  the  special  form  shown. 

The  inclination  of  the  conduit  is  1  foot  per  mile,  and  the  flow  80,000,000 
gallons  per  24  hours  when  full  or  5  feet  above  center  of  invert.  The  maximum 
flow  takes  place  when  the  depth  of  water  is  7'  2",  the  delivery  then  being  109,- 
000,000  gallons. 

In  large  works,  where  there  is  considerable  length  of  conduit,  receiving 
reservoirs,  within  or  near  the  limits  of  the  city,  are  necessary  as  a  precaution 
to  guard  against  accidents  which  might  happen  to  conduit  or  dam,  and  cut  off 
the  supply,  and  also  as  a  sort  of  balance  against  unequal  or  intermittent  draught 
among  the  consumers.  The  size  of  these  reservoirs  must  depend  on  the  neces- 
sities of  the  case,  and  on  the  facilities  for  construction.  The  capacity  of  the 
Kidgewood  reservoir,  at  Brooklyn,  is  161,000,000  gallons  when  full ;  of  the 
new  Croton  reservoir,  about  1,000,000,000  gallons.  Both  these  reservoirs  are 
made  double — that  is,  in  two  compartments. 


FIG.  860. 

Fig.  860  is  a  section  of  the  division-bank  of  the  new  Croton  reservoir.  It 
is  made  of  earth,  with  a  puddled  ditch  in  the  center,  and  slopes  protected  by 
rock-paving. 

A  few  extracts  from  the  specification  will  explain  the  general  construction 
of  the  reservoir  : 

"  The  reservoir  will  be  formed  by  an  exterior  bank  forming  the  outer  sides  of  the 
basin.  There  will  be  a  division-bank,  dividing  the  reservoir  into  two  basins.  All  the 
banks  will  have  the  inner  and  outer  slopes  of  1£  base  to  1  perpendicular.  All  the  inner 
or  water-slopes  will  be  covered  with  8"  of  broken  stone,  on  which  will  be  placed  the 
stone  pavement,  H  feet  thick.  The  outer  slopes  will  be  covered  with  soil  1  foot  thick. 
The  banks,  when  finished,  to  be  15  feet  on  top,  exclusive  of  the  soil  on  the  outer  slope. 
The  top  of  the  outer  bank  to  be  4  feet  above  water-line ;  the  top  of  the  division-bank  to 
be  3  feet  below  water-line.  In  the  center  of  all  the  banks  a  puddle-bank  will  be  built,  ex- 
tending from  the  rock  to  the  paving  in  the  division-bank,  and  to  within  2  feet  of  the  top 
of  the  outer  bank.  It  will  be  6'  2"  wide  at  top  in  division-bank,  and  14'  wide  at  top  in 


ENGINEERING  DRAWING. 


395 


exterior  bank,  and  16'  wide  at  a  plane  38'  below  top  of  exterior  bank.  In  the  middle  of 
the  division-bank  there  will  be  built  a  brick  wall,*  laid  in  cement-mortar,  4'  high,  20"  wide, 
the  top  of  the  wall  to  be  connected  with  the  bottom  of  the  stone  pavement ;  8"  thickness 
of  concrete  is  to  be  laid  on  the  top  of  the  bank,  on  each  side  of,  and  connected  with,  this 
wall.  On  the  pavement  18"  thick  will  be  laid  in  concrete.  The  slope-wall  on  each  side 
of  the  division-bank,  10'  in  width, to  be  laid  in  cement. 

"  Puddle-ditches  are  to  be  excavated  to  the  rock  under  the  center  of  all  embankments 
where  the  rock  is  not  over  46'  below  top  of  exterior  bank.  Where  the  rock  is  more  than 
46',  two  rows  of  sheet-piling  are  to  be  driven  to  the  rock,  16'  apart,  and  the  material  be- 
tween them  excavated,  so  as  to  remove  all  soil,  muck,  or  vegetable  matter.  Sheet-piling 
will  be  formed  of  spruce  or  pine  plank,  6"  thick,  tongued  and  grooved;  the  tongue  and 
groove  to  be  1$"  x  1".  The  earth  within  the  working-lines  of  interior  slopes  will  be  ex- 
cavated to  the  depth  of  40'  below  top  of  exterior  bank,  rock  36'.  The  puddle-ditch  will 
be  formed  of  clay,  gravel,  sand,  or  earth,  or  such  admixture  of  these  materials,  or  any  of 
them,  as  the  engineer  may  direct,  to  be  laid  in  layers  of  not  more  than  6",  well  mixed  with 
water,  and  worked  with  spades  by  'cutting  through  vertically,  in  two  courses  at  right 
angles  with  each  other;  the  courses  to  be  1"  apart,  and  each  spading  to  extend  2"  into  the 
lower  course  or  bed.  Whenever  the  work  is  suspended,  the  puddle  must  be  covered  with 
boards  or  earth  to  prevent  cracking,  and,  whenever  cracks  do  occur  in  the  puddle,  those 
parts  must  be  removed  and  reworked.  The  puddle  will  extend  to  all  the  masonry  and 
pipes,  and  along  and  around  it  and  them  as  the  engineer  may  direct. 

"The  embankments  will  be  formed  in  layers  of  not  more  than  6",  well  packed  by 
carting  and  rolling,  and,  in  such  places  as  the  rollers  can  not  be  effectually  used,  by  ram- 
ming. The  embankments  will  be  worked  to  their  full  width  as  they  rise  in  height,  and  not 
more  than  2'  in  advance  of  the  puddle.  The  interior  slopes  of  all  the  banks  will  be 
covered  with  8"  thickness  of  stone,  broken  to  pass  through  a  2"  ring.  On  this  will  be 
laid  the  paving,  18"  in  thickness,  of  a  single  course  of  stones  set  on  edge  at  right  angles 
with  the  slope,  laid  dry,  and  well  wedged  with  pinners." 


FIG.  861. 


FIG.  862. 


FIG.  863. 


FIG.  864. 


Distribution. — Figs.  861  to  865  are  sections  of  the  spigot  and  faucet  ends 
of  some  of  the  pipes  of  the  city  of  Brooklyn.  Of  these  pipes  there  were  two 
classes,  A  and  B.  The  A  pipes  were  designed  for  positions  subject  to  an  ex- 


*  This  wall  was  formed  of  concrete. 


396 


ENGINEERING  DRAWING. 


treme  head  of  120',  the  B  pipes  for  positions  below  this  level,  subject  to  a 
head  of  from  120  to  170  feet. 

The  thicknesses  of  these  pipes  is  greater  than  those  which  now  obtain  in 
practice.  The  following  table,  made  from  the  average  of  formulas  and  of  the 
dimensions  in  use  in  different  cities,  may  be  considered  safe  for  a  static  press- 
ure of  100  Ibs.,  or  231  feet.  But  pipes  should  be  tested  at  the  manufactories 
to  three  times  this  pressure.  The  weights  given  are  the  pipes  as  delivered  in 
lengths  of  12'  or  12'  5";  as  laid,  the  laps  are  5",  and  for  running  feet  about 
4  per  cent  should  be  added  to  the  table-weights  : 


LEAD   JOINT. 

TV 

Th.ick.n6ss 

Weight 

Depth. 

Weight. 

In. 

Lbs. 

4 

•42 

18 

1* 

4i 

6 

•47 

30 

1| 

6i 

8 

•52 

44 

1* 

8* 

10 

•58 

60 

If 

10i 

12 

•63 

78 

2 

13 

16 

•73 

120 

2i 

24^- 

20 

•83 

170 

g£ 

31 

24 

•94 

228 

2f 

38 

30 

1-10 

330 

2* 

57 

36 

1-24 

450 

2^ 

30 

48 

1-44 

700 

H 

111 

The  smallest  water-pipe  laid  in  large  cities  now  is  the  6";  the  other  sizes 
given  in  the  table  are  in  common  use  and  are  found  in  stock,  except  the  10", 

which  can  be  obtained  by  order. 
In  laying,  a  hemp  gasket  is  forced 
down  to  the  lower  end  of  the  bell 
to  prevent  the  molten  lead  escap- 
ing into  the  pipe.  The  end  of  the 
pipe  is  then  stopped  by  the  clay 
roll,  or  a  rope  covered  with  clay, 
or  clay  alone,  and  the  melted  lead 
poured  in  through  an  aperture  or 
gate  at  the  top.  After  cooling,  the 
lead  is  calked  or  compacted  in  the 
joint. 

Specials. — All  parts  of  a  main 
except  the  straight  pipes  are  called 
special  castings. 

Fig.  866  is  a  12"  X  8"  4-way 
branch,  shown  full  and  in  section. 

J?IG.   oob. 

diagonally.     The  horns  on  the  4" 

branch  are  for  the  straps  which  hold  in  the  plug,  or  cap,  or  a  connected  short 
or  curved  pipe.     The  4-way  branches  are  often  called  crosses,  and  the  3- way 


ENGINEERING  DRAWING. 


397 


T's,  or  single  branches.     The  branches  may  be  of  any  appropriate  size.     In 
ordering,  designate  diameter  of  main  pipe  first,  and  then  that  of  the  branches. 
It  is  very  common  in  these  pipes  to  make  all  the  ends  bell  ends — it  saves 
sleeves  when  pipes  are  cut,  as  they  usually  are  at  street 
intersections. 

Fig.  867  is  a  section  of  a  sleeve  for  uniting  cut 
pipes  or  uncut  spigot-ends  ;  a  kind  of  double  hub  is 
often  used  for  the  former.     Some- 
times sleeves  are  made  in  halves, 
and  bolted  together. 

Fig.  868  is  the  section  of  a  re- 
ducer for  the  connection  of  pipes 
of  unequal  diameters. 

Fig.  869  'is  the  section  of  a 
bend;  the  horns  on  the  outer  cir- 
cle are  for  straps  between  the  pipes, 
as  the  pressure  is  unbalanced. 

Fig.  870  is  a  section  of  the  con- 
nection of  two  wrought-iron  pipes 
by  a  bell  riveted  to  the  end  of  one,  and  a  fillet  or  ring 
to  the  end  of  the  other.  FIG.  868. 

House-services  are  usually  through  lead  pipes ;  the 
taps  allowed  on  the  mains  for  house-connections  being  usually  from  -J"  to  f ". 


FIG. 


FIG.  869. 

From  the  specifications  of  "  Cast-iron  Distribution-Pipes  and  Pipe-Mains, 
with  their  Branches,"  etc.,  Brooklyn,  L.  I. : 

"  All  pipes  of  20"  diameter  and  upward  to  be  formed  so 
as  to  give  a  lead  joint  of  not  less  than  f"  in  thickness  all 
round,  and  not  more  than  Ty ;  those  of  12"  diameter  and 
under,  not  exceeding  |",  and  not  less  than  Ty.  The  straight 
pipes  of  12"  diameter  and  upward  shall  be  cast  in  dry  sand 
molds,  vertically.  The  smaller  pipes  may  be  cast  at  an  angle 
with  the  horizon  of  not  less  than  12°.  The  pipes  shall  be 
free  from  scoria,  sand-holes,  air-bubbles,  cold-short  cracks, 
and  other  defects  or  imperfections ;  they  shall  be  truly  cylindrical  in  the  bore,  straight 


FIG.  870. 


398  ENGINEERING  DRAWING. 

in  the  axes  of  the  straight  pipes,  and  true  to  the  required  curvature  or  form  in  the  axes 
of  the  other  pipes;  they  shall  be  internally  of  the  full  specified  diameters,  and  have  their 
inner  and  outer  surfaces  concentric.  No  plugging  or  filling  will  be  allowed.  They  shall 
be  perfectly  fettled  and  cleaned  ;  no  lumps  or  rough  places  shall  be  left  in  the  barrels  or 
sockets.  No  pipes  will  be  received  which  are  defective  in  joint-room.  The  spigot  ends 
of  all  the  branches  to  have  lugs  or  horns  cast  on  each.  Every  pipe-branch  and  casting 
shall  pass  a  careful  hammer-inspection,  and  shall  be  subject  thereafter  to  a  proof  by  water- 
pressure  of  300  Ibs.  to  the  square  inch  for  all  pipes  30"  in  diameter  and  under,  and 
250  Ibs.  per  square  inch  for  all  pipe-mains  exceeding  30"  diameter.  Each  pipe,  while 
under  the  required  pressure,  shall  be  rapped  with  a  hand-hammer  from  end  to  end,  to 
discover  whether  any  defects  have  been  overlooked.  The  pipes  shall  be  carefully  coated 
inside  and  outside  with  coal-pitch  and  oil,  according  to  Dr.  R.  A.  Smith's  process,  as 
follows: 

"Every  pipe  must  be  thoroughly  dressed  and  made  clean  from  sand  and  free  from 
rust.  If  the  pipe  can  not  be  dipped  presently  after  feeing  cleansed,  the  surface  must  be 
oiled  with  linseed-oil  to  preserve  it  until  it  is  ready  to  be  dipped ;  no  pipe  to  be  dipped 
after  rust  has  set  in.  The  coal-tar  pitch  is  made  from  coal-tar,  distilled  until  the  naphtha 
is  entirely  removed  and  the  material  deodorized.  The  mixture  of  five  or  six  per  cent  of 
linseed-oil  is  recommended  by  Dr.  Smith.  Pitch,  which  becomes  hard  and  brittle  when 
cold,  will  not  answer.  The  pitch  must  be  heated  to  300°  Fahr.,  and  maintained  at  this 
temperature  during  the  time  of  dipping.  Every  pipe  to  attain  this  temperature  before 
being  removed  from  the  vessel  of  hot  pitch.  It  may  then  be  slowly  removed  and  laid 
upon  skids  to  drip." 

Sewers. — For  the  removal  of  waste  water  from  houses  and  rainfall,  sewers 
are  very  convenient  in  towns  and  cities,  even  before  the  construction  of  water- 
works ;  but,  after  the  introduction  of  a  liberal  and  regular  supply  of  water, 
sewers  are  indispensable.  The  ruling  principle  in  the  establishment  of  sewer- 
age-works is,  that  each  day's  sewage  of  each  street  and  of  each  dwelling  should 
be  removed  from  the  limits  of  city  and  town,  as  far  as  practicable,  on  the  day 
of  its  production,  that  it  should  pass  off  before  decomposition  begins,  and  that 
it  should  not  be  allowed  to  settle  and  fester  in  the  sewers,  producing  those  nox- 
ious gases  so  prejudicial  to  health.  To  attain  this  end,  the  refuse  fluids  must 
be  sufficient  in  quantity  to  float  and  carry  off  the  heavier  matters  of  sewage. 

There  has  been  considerable  discussion  of  late  whether  sewage  and  rainfall 
should  be  carried  off  by  a  single  system  of  pipes.  This  must  depend  largely  on 
the  location,  economy  of  construction,  and  the  financial  ability  to  carry  out 
the  design.  If  the  rainfall  can  be  provided  for  by  street  gutters,  the  pipes  for 
the  conveyance  of  house-waste  may  be  very  small. 

If  the  rate  of  inclination  of  a  sewer  be  not  less  than  1  in  440,  the  experience 
of  Brooklyn,  and  other  cities  equally  well  supplied  with  water,  shows  that  the 
fluid  of  domestic  sewage  is  sufficient  to  carry  off  all  the  heavier  matters,  and 
keep  the  sewers  free  and  clean,  provided  the  form  is  such  as  to  concentrate  as 
much  as  possible  the  sewage  waters.  Less  inclination  than  1  in  440  will  require 
some  means  of  flushing.  In  the  Brooklyn  system  of  sewers,  adopted  on  the 
report  and  plans  of  Colonel  J.  W.  Adams,  the  principle  of  construction  has 
been,  to  make  the  sewers  as  small  as  the  service  required  of  them  will  admit, 
to  maintain  as  much  velocity  of  flow  as  possible,  so  that  nothing  may  be  de- 
posited ;  and  without  any  provision  for  a  man  entering  and  passing  through 
the  sewer,  which  has  been  found  by  experience  unnecessary. 


ENGINEERING  DRAWING. 


399 


The  value  of  sewers  depends  on  the  cor- 
rectness of  their  lines,  uniformity  of  de- 
scent, and  smoothness  of  interior  surface. 
The  pipes  used  in  Brooklyn  have  generally 
been  strong  glazed  earthenware  pipes,  of 
12",  15",  and  18"  diameter.  Many  cement- 
pipes  have  also  been  used,  and,  in  such 
situations  as  required  great  capacity,  brick 
sewers  were  used,  the  leading  forms  of  which 
are  egg-shaped,  as  in  Fig.  871,  of  which  the 
dimensions  are  as  follow  :  R  (as  in  table) 
the  longest  diameter  D,  and  the  longest 
radius  R',  each  3  times  R,  and  R"  £  R. 


FIG.  871. 


AEEA. 

K. 

D. 

fiO" 

diameter  circular        .  .                                 .  . 

24-8 

74-4 

-18" 

u                 u 

19-8 

59'5 

Sfi" 

u                 u 

14-9 

44'7 

94" 

u                 a 

9-9 

29'8 

Thickness  of  brickwork,  8"  ;  boards  shown  at  bottom  only  used  in  cases  of 
soft  earth  for  convenience  of  construction.  For  area  of  egg-shaped  sewer  of 
above  section,  multiply  R2  by  4 -6. 

In  some  locations  the  depth  did  not  admit  of  the  egg-shaped  section.  A 
circular  form  of  6  feet  diameter  was  adopted  for  the  Union  Avenue  sewer,  and 
one  of  a  section  similar  to  the  main  conduit  of  the  water-works,  10  feet  in 
width  and  9  feet  high,  in  the  clear,  for  Kent  Avenue. 

Fig.  872  is  a  section  of  the  largest  Washington  sewer.  The  bottom  course 
of  the  large  sewers,  or  where  exposed  to  a  strong  current,  are  of  stone ;  the 
ring-courses,  of  brick,  are  3  for  the  13-foot  sewers  and  2  for  the  7-foot. 

Man-holes  are  built  along  the  line  of  sewers,  at  a  distance  of  from  100  to 
150  feet  apart,  to  give  access  to  the  sewers  for  purposes  of  inspection  and  re- 
moval of  deposit. 

Figs.  873  and  874  are  section  and  plan  of  the  man-hole  at  present  used  by 
the  Croton  Sewer  Department.  It  consists  of  a  funnel-shaped  brick  well,  oval 
at  the  bottom,  4'  X  3',  circular  at  top,  2'  diameter,  curbed  with  cast-iron  frame 
and  covered  by  cast-iron  plate.  Side-walls,  8"  thick,  through  which  the  pipe- 
sewers  pass  at  the  bottom  of  the  well.  Across  the  open  space  the  sewer  is  formed 
in  brick,  whose  bottom  section  corresponds  to  that  of  pipe,  side-walls  carried 
up  perpendicular  to  top  of  sewer  ;  the  flat  spaces  at  the  sides  of  sewer  are 
flagged.  The  top  of  the  sewer  is  a  heavy  cast  -  iron  frame,  fitted  with  a 
strong  cover,  which  may  or  may  not  be  perforated,  for  ventilation.  In  the 
figure  the  main  sewer  is  12"  pipe,  with  a  12"  branch  entering  at  an  acute 
angle,  as  all  branches  and  connections  with  a  sewer  should.  The  short  lines 
on  the  left  vertical  wall  represent  sections  of  f\  staples,  built  in  to  serve  for  a 
ladder. 


400 


ENGINEERING  DRAWING. 


Wherever  necessary,  from  the  slope  and  conformation 
of  the  ground,  to  remove  the  surface  or  rain  water  direct- 


ly  from  the  street-gutters  into  the 
3g  sewers,  catch-basins  are  placed  gen- 
H  erally  at  the  corners  of  streets. 

Figs.  875  and  876  are  section  and 

plan  of  the  Croton  sewer  catch-basins. 


ENGINEERING  DRAWING. 


401 


on  a  scale  of  y  =  1  foot.     The  intention  of  the  catch-basin  is  to  receive  the 

street  washings,  retain  the  heaviest  portion  in  the  basin,  and  let  the  liquid 

escape  into  the  sewer.     The  basin  in  the  figure  is  rectangular  in  plan,  with  a 

semicircular  end,  3'  8"  in  width 

by  5'  I"  long ;  bottom  of  flag  and 

side-walls  of  brick,  12"  thick.    It 

will  be  observed  that  a  piece  of 

flag  is  built  into  the  side-walls 

from  the   top,  extending  about 

lialf-way   to   the   bottom ;    this 

divides  the   upper  part  of  the 

basin  into  two  parts  ;  the  sewer 

enters  the  basin  three  feet  above 

bottom  flag  ;    the  dividing  flag 

comes  to  within   2'  6"  ;   before 

any  water  can  flow  out  through 

the  sewer-pipe  this  flag  must  be 

submerged  6"  ;    a   trap   is  thus 

formed,  which  cuts  off  any  smell 

from  the  sewer  escaping  into  the 

street.     This  trap  is  sometimes 

made  of  a  cast-iron  elbow,  turned 

down  and  bolted  to  the  sewer-  Fl°-  8^5- 

pipe  in  the  wall.     The  water  is 

received  into  the  basin  through  the  channel  0,  which  is  curbed  with  granite, 

and  protected  by  a  grating.     The  coping  (b)  is  of  granite,  and  forms  a  portion 

of  the  sidewalk  ;  through  this  there  is  a  man-hole  cut,  16"  diameter,  for  access 

to  the  basin,  for  removal  of  the 
deposit ;  it  is  covered  by  a  strong 
cast-iron  plate. 

Gas- Supply. — Next  in  impor- 
tance to  the  necessities  of  a  city  or 
town  for  water-supply  and  sewer- 
age, is  the  luxury  of  gas-supply. 
The  gas-works  should  always  be 
placed  remote  from  the  thickly- 
populated  part  of  a  city,  for  under 
the  best  regulations  some  gas  will 
escape  in  the  manufacture,  offen- 
sive and  deleterious.  They  should 

be  placed  at  the  lowest  level,  for,  gas  being  light,  readily  rises,  and  the  portions 

of  the  city  below  the  works  are  supplied  at  less  pressure  than  those  above. 

Gas-mains,  like  those  for  water,  are  of  cast-iron,  and  put  together  in  the  same 

way  ;  but,  as  they  have  to  resist  no  pressure  beyond  that  of  the  earth  in  which 

they  are  buried,  they  are  never  made  of  as  great  thickness  as  those  of  water- 
pipes,  but  drips  must  be  provided,  and  the  pipe  laid  with  such  inclination  to 

them  that  the  condensed  tar  may  be  received  in  them  and  pumped  out. 


FIG.  876. 


402 


ENGINEERING  DRAWING. 


WEIGHT   OF   GAS-PIPES   PER   RUNNING  FOOT. 

8" 12  Ibs. 

4" 16    " 

6"..  .  27    " 


40 


10" 50  Ibs. 

12" 62    " 

16"..  .    103    " 


150 


FIG.  877. 


Roads. — Under  this  general  term  are  included  all  routes  of  land-travel  ;  but 
the  term  "streets"  is  applied  mostly  to  city,  town,  and  suburban  roads,  while 
"roads"  and  "highways"  are  applied  to  those  of  the  country.  By  an  "ave- 
nue" is  generally  understood  a  wide  street.  In  New  York  all  the  streets  run- 
ning north  and  south  are  called  avenues,  and  those  at  right  angles,  streets,  and 
the  term  boulevard  to  very  wide  avenues  in  which  there  are  rows  of  trees.  The 

terms  street  and  avenue,  as  laid  out, 
are  the  established  bounds  within 
which  no  buildings  may  be  erected. 
The  street,  therefore,  technically  in- 
cludes the  street  or  traveled  way  for 
carriages,  and  the  sidewalks  and  front 
areas.  New  York  streets  above  Four- 
teenth Street  are  60  and  100  feet  wide, 

avenues  100  feet,  of  which  the  carriage-way  occupies  one  half,  and  the  sidewalks 
and  area  one  quarter  on  each  side.  The  space  occupied  by  areas,  is  from  5  to 
8  feet,  which  may  be  inclosed  by  iron  fence,  but  can  not  be  included  within 
the  building  above  the  level  of  sidewalk.  The  stoop-line  extends  into  the 
sidewalk  beyond  the  area-line  some  1'  to  18",  fixing  the  limit  for  the  first  step 
and  newel  to  a  high  stoop  or  platform.  The  boulevard  in  the  old  line  of 
upper  Broadway  and  the  Bloomingdale  Eoad  is  150  feet  wide,  of  which  100 
feet  are  to  be  carriage-way,  and  25  feet  on  each  side  for  sidewalk  and  area,  the 
latter  not  to  exceed  7  feet ;  one  row  of  trees  to  be  set  within  the  sidewalk, 
about  2  feet  from  the  curb. 

In  Paris,  there  is  no  area ;  the  sidewalk  comes  up  to  the  house  or  street-line, 
and  there  is  a  space  for  trees  between  sidewalk  and  street-curb.  This  space  is 
available  for  pedestrians,  a  part  being  a  gravel,  asphalt,  or  flagged  walk.  The 
following  are  the  dimensions  according  to  the  law  of  June  5,  1856  : 


Entire  width  of 
boulevard  and 
avenues. 

Width  of 
carriage-way. 

Width  of 
sidewalk. 

Width  for 
trees. 

Rows  of 
trees. 

DISTANCE  OF  EOW  FROM 

Street-line. 

Street-curb. 

Metres. 

Metres. 

Metres. 

Metres. 

Metres. 

Metres. 

26  to  28 

12 

1 

5'5  to  6-5 

1-5 

30  "  34 

14 

1 

6-5  "  8-5 

1-5 

36  "  38 

12  to  13 

3-5 

8'  to  8-5 

2 

5'     "  5-5 

1-5 

40 

14 

3-6 

9-5 

2 

6'5 

1-5 

1  metre  =  3'281  feet. 


The  foot-walks  in  this  city  and  vicinity  are  generally  formed  of  flags,  or 
what  is  here  termed  blue-stone,  laid  on  a  bed  of  sand  or  cement-mortar.  The 
flags  are  from  2"  to  4"  thick.  In  the  more  important  streets  the  upper  surface 


ENGINEERING  DRAWING. 


403 


is  axed,  the  quality  of  the  stone  selected,  and  the  sidewalk  often  covered  by  a 
single  width  of  stone.  Brick  are  often  used  in  towns,  or  places  where  good 
flag  can  not  be  readily  obtained,  usually  laid  flatways  on  a  sand-bed.  Granite 


Carriage-way. 


FIG.  8V8. 

is  very  often  employed  in  business  streets,  in  lengths  the  full  width  of  the  side- 
walk, and  about  1'  in  thickness,  the  inner  ends  resting  on  an  iron  girder, 
and  the  outer  on  the  vault- wall,  forming  in  this  way  a  roof  for  the  vault  and  the 
outer  ends  a  curb  for  the  street. 

Curls  here  are  generally  of  flag,  about  4"  thick  by  20"  deep,  extending 
10*  above  the  gutter-stone ;  but,  where  the  street  is  nearly  level,  and  the 
gutter-stones  have  to  be 

raised   to    give   sufficient  Curb-      Sidewalk, 

descent  for  the  flow  of  the 
water,  the  curbs,  in  ex- 
treme cases,  are  not  more 
than  4"  exposed.  When 
sidewalks  are  stone  of 
large  dimensions,  they  ex- 
tend over  the  curbs.  The  FIG.  879. 
gutter-stones  are  from  12" 

to  15"  in  width,  and  from  3"  to  5"  in  thickness,  laid  close,  and  bedded  in 
cement.  The  bridge  or  crossing-stones  are  of  blue-stone  or  granite,  from  2' 
to  15"  wide,  and  not  less  than  5"  thick,  laid  in  double  rows. 

Carriage-way. — Streets  and  avenues  were  formerly  paved  entirely  of  cobble- 
stone, and,  if  selected  so  as  to  be  of  a  uniform  size  and  shape,  and  properly 
bedded  in  sand  and  well  rammed,  they  formed  a  cheap  and  very  fair  roadway ; 
but  the  cubical  block-stone  pavement  of  trap  or  granite,  often  called  the  Bel- 
gian, is  in  every  way  to  be  preferred.  Mr.  Kneass,  the  engineer  of  the  city  of 
Philadelphia,  recommends  : 

"  That  the  blocks  should  not  exceed  3"  in  width,  6"  in  depth,  nor  8"  in  length ;  that, 
as  to  depth,  they  should  be  uniform  within  J",  and  in  length  he  not  less  than  6".  For 
foundations  the  material  should  he  taken  out  to  a  depth  of  20"  below  the  proposed  surface 
of  paving,  and  to  be  made  to  accurately  conform  to  shape  of  finished  road.  After  being 
compactly  rolled  with  a  heavy  roller,  it  should  have  a  covering  of  clean  anthracite  coal- 
ashes  placed  upon  it  to  a  depth  of  10",  laid  on  in  two  layers,  each  well  rolled;  the  ashes 
to  be  scrupulously  clean — i.  e.,  free  from  any  organic  matter.  Upon  this  should  be  laid  a 
bed  of  clean  gravel,  4"  in  depth,  and  rolled;  upon  which  again  should  be  a  layer  of  sand, 
clean  and  sharp,  or  fine-screened  gravel,  in  which  to  set  and  bed  the  stone  blocks.  Each 
layer  of  ashes  and  gravel  should  in  surface  conform  to  the  outline  intended  for  the  surface 


404  ENGINEERING  DRAWING. 

of  the  stone.  The  stone  should  be  carefully  assorted,  so  that,  when  laid  across  the  street, 
the  joint-lines  may  be  straight ;  and  each  stone  should  he  set  on  its  ~bed  fair  and  square,  so 
that  no  edge  shall  extend  above  the  general  level  of  the  surface,  and  the  surface  of  each 
stone  shall  be  an  extension  of  that  lying  next  to  it.  The  joints  I  would  not  make  smaller 
than  £",  to  be  filled  with  sand  and  grouted  with  liquid  lime.  Before  grouting,  the  entire 
surface  should  be  rammed  until  no  impression  can  he  made  on  it" 

New  York  pavements  are  usually  laid  of  granite  or  trap-blocks,  4"  wide, 
6"  deep,  and  8"  to  12"  long,  set  in  sand  simply,  or  on  a  concrete  base.  In 
London  the  usual  practice  is  to  set  their  blocks  3"  wide  by  9"  deep,  and  from 
6"  to  12"  long,  on  a  bed  of  gravel,  with  a  base  of  broken  granite  12"  deep. 

Wooden  pavements  of  various  kinds  have  been  tried.  The  "Nicholson" 
consists  of  pieces  of  3"  plank,  6"  long,  set  on  a  board  base  supported  by  a 
sand-bed.  The  plank  is  set  on  end  in  lines  perpendicular  to  line  of  street, 
with  a  strip  of  board  I"  wide  between  the  rows,  nailed  to  the  blocks  ;  the  top 
of  strip  being  some  2"  to  3"  below  top  of  plank.  Boiled  coal-tar  is  used  freely 
while  setting  the  bloqks,  and  is  poured  into  the  interstices  ;  the  I"  joint  is  filled 
with  gravel,  wet  with  tar,  and  well  rammed.  Instead  of  plank,  blocks  of  wood, 
sawed  from  trunks  or  limbs,  from  4"  to  9"  diameter,  with  the  bark  removed, 
are  set  on  a  board  or  plank  base,  with  the  interstices  filled  with  gravel,  into 
which  coal-tar  or  melted  asphalt  is  poured,  and  the  top  covered  with  gravel 
and  thoroughly  rolled  into  the  wood,  so  that  the  wear  is  on  the  gravel.  In 
all  cases,  although  more  expensive,  it  is  better  to  make  a  concrete  base. 

Of  late  years,  asphalt  has  been  used  abroad  to  a  very  great  extent,  both 
for  foot  and  carriage  ways.  The  carriage-ways  are  composed  of  a  layer  of 
asphalt,  from  1£"  to  2"  thick,  on  a  bed  of  concrete,  or  on  a  worn  Macadam  road, 
over  which  is  spread  a  thin  coat  of  cement.  The  cement  having  become  dry, 
the  asphaltic  rock,  reduced  to  a  powder,  is  spread  over  the  surface  to  a  depth 
of  about  *40  per  cent  more  than  the  finished  thickness  ;  it  is  then  rammed  with 
rammers  warmed  by  portable  furnaces,  beginning  gradually,  and  increasing  the 
force  of  the  blows  as  the  work  approaches  completion.  For  a  footway  the  same 
concrete  bed  is  used,  and  the  layer  of  asphalt  is  about  |".  Walks  and  roads 
have  been  constructed  in  this  country  with  an  artificial  asphalt,  prepared  from 
coal-tar  mixed  with  gravel. 

Pavements  of  mineral  asphalt  have  also  been  laid  in  many  of  our  cities.  In 
Washington,  the  asphalt  pavement  consists  of  6"  of  hydraulic  cement  concrete 
and  a  wearing  surface  of  bituminous  mastic  laid  in  two  coats  respectively  -J"  and 
2"  thick  when  compressed.  The  mastic  is  composed  of  the  following  parts  by 
weight : 

Asphaltic  cement  (refined  Trinidad  asphalt)  100  parts,  petroleum  oil  20  parts.     15  to    18 

Limestone  powder 15  to    17 

Sand  . .  TO  to    65 


100  to  100 

Roads  and  Highways. — Macadam  was  the  first  to  reduce  the  construction  of 
broken-stone  roads  to  a  science,  and  has  given  his  name,  in  this  country,  to  all 
this  class  of  roads.  He  says  that  "  the  whole  science  of  artificial  road-making 
sonsists  in  making  a  dry,  solid  path  on  the  natural  soil,  and  then  keeping  it 
dry  by  a  durable  water-proof  covering."  The  road-bed,  having  been  thoroughly 


ENGINEERING  DRAWING. 


405 


drained,  must  be  properly  shaped,  and  sloped  each  way  from  the  center,  to  dis- 
charge any  water  that  may  penetrate  to  it.  Upon  this  bed  a  coating  of  3"  of 
clean  broken  stone,  free  from  earth,  is  to  be  spread  on  a  dry  day.  This  is  then 
to  be  rolled,  or  worked  by  travel  till  it  becomes  almost  consolidated  ;  a  second 
3"-layer  is  then  added,  wet  down  so  as  to  unite  more  readily  with  the  first ;  this 
is  then  rolled,  or  worked,  and  a  third  and  fourth  layer,  if  necessary,  added. 
Macadam's  standard  for  stone  was  6  ounces  for  the  maximum  weight,  corre- 
sponding to  a  cube  of  1J",  or  such  as  would  pass  in  any  direction  through  a  %y 
ring.  The  Telford  road  is  of  broken  stone,  supported  on  a  bottom  course  or 
layer  of  stone  set  by  hand  in  the  form  of  a  close,  firm  pavement. 

At  the  New  York  Central  Park,  after  trials  of  the  Macadam  and  Telford  roads, 
the  gravel-road  (of  which  Fig.  880  is  a  cross-section  of  one  half)  was  adopted,  as 
being,  according  to  the  statement  of  their  engineer,  Mr.  Grant,  "the  easiest  and 


FIG. 


most  agreeable  kind  of  road  for  both  carriages  and  horses  ;  that  it  is  the  cheapest 
at  first  cost,  and  can  be  kept  in  repair  at  an  equal  if  not  less  cost  than  any  other 
equally  satisfactory  road."  This  road  consists  of  a  layer  of  rubble-stones,  about 
7"  thick,  on  a  well-rolled  or  packed  bed,  with  a  covering  of  5"  of  clean  gravel. 
C  is  the  catch-basin  for  the  reception  of  water  and  deposit  of  silt  from  the 
gutters  ;  S  is  the  main  sewer  or  drain,  and  s  a  sewer-pipe  leading  to  catch-basin 
on  opposite  side  of  the  road.  In  wider  roads  each  side  has  its  own  main  drain, 
and  there  is  no  cross-pipe  s.  The  road-bed  was  drained  by  drain-tiles  of  from 
iy  to  4"-bore,  at  a  depth  of  3'  to  3J  below  the  surface.  The  maximum  grade 
of  the  Park  roads  is  1  in  20.  The  grades  of  the  streets  of  Paris  vary  from  1  in 
20  to  1  in  200.  The  best  grade  is  from  1  in  50  to  1  in  100  ;  this  gives  ample 
descent  for  the  flow  of  water  in  the  gutters.  Many  of  our  street-gutters  have 
a  pitch  not  exceeding  1  foot  in  the  width  of  a  block,  or  200  feet. 

The  grade  of  a  road  is  described  as  1  in  so  many ;  so  many  feet  to  the  mile, 
or  such  an  angle  with  the  horizon. 


Inclination. 

Feet  per  mile. 

Angle. 

Inclination. 

Feet  per  mile. 

Angle. 

1  in  10 

528 

5°  43' 

in    30 

176 

1°  55' 

1  "  11 

462 

5° 

"     40 

132 

1°  26' 

1  "  14 

369 

4° 

"     50 

106 

1°    9' 

1  "  20 

264 

2°  52' 

"     57 

92 

1° 

1  "  29 

184 

2° 

"  100 

53 

35" 

406 


ENGINEERING  DRAWING. 


The  best  transverse  profile  for  a  road  on  nearly  level  ground  is  that  formed 
by  two  inclined  planes,  meeting  in  the  center,  and  having  the  angle  rounded. 
The  degree  of  inclination  depends  somewhat  on  the  surface  of  the  road.  A 
medium  for  broken-stone  roads  is  about  -J"  in  1',  or  1  in  24  ;  but  Telford,  on  the 
Holyhead  Road,  adopted  1  in  30  ;  and  Macadam,  1  in  36,  and  even  1  in  60. 
For  paved  streets  in  Paris,  a  crown  of  -fa  of  the  width  is  adopted,  and  for 
Macadamized,  -3-^.  The  inclination  of  sidewalks  should  not  exceed  f"  in  1 
foot,  and,  when  composed  of  granite,  the  surface  should  be  roughened. 

The  necessity  of  a  well-drained  road-bed  is  as  important  beneath  rails  as  on 
a  highway.  The  cuts  should  be  excavated  to  a  depth  of  at  least  2  feet  below 
grade,  with  ditches  at  the  sides  still  deeper,  for  the  discharge  of  water.  The 
embankments  should  not  be  brought  within  2  feet  of  grade  ;  this  depth  to  be 
left  in  cut  and  on  embankment  for  the  reception  of  ballast.  The  best  ballast 
is  Macadam  stone,  on  which  the  cross-ties  are  to  be  bedded,  and  finer  broken 
stone  packed  between  them.  Good  coarse  gravel  makes  very  good  ballast ;  but 
sand,  although  affording  filtration  for  the  water,  is  easily  disturbed  by  the  pas- 
sage of  the  trains,  raising  a  dust,  an  annoyance  to  travelers,  and  an  injury  to 
the  rolling-stock  by  getting  into  boxes  and  bearings.  The  average  length  of 
sleepers  on  the  4.8J-  gauge  railways  is  about  8  feet ;  bearing  surface,  7";  dis- 
tance between  centers,  2  feet  to  2'  6",  except  at  joints,  where  they  are  as  close 
to  each  other  as  the  necessity  of  tamping  beneath  them  will  admit.  Average 
width  of  New  York  railways  of  same  gauge  as  above,  for  single  lines,  in  cuts 
18',  banks  13' ;  for  double  lines  cuts  31',  banks  26f. 


U ix* .j 

....AW. ,.'U'.':A'.J 


FIG.  881. 


CROSS    SECTION    GRAVEL  BALLAST. 
FIG.  882. 


Figs.  881  and  882  are  two  standard  sections  of  the  permanent  way  of  the 
Pennsylvania  Railroad,  in  which  the  width  of  cuts  and  top  of  embankments 
are  the  same,  31'  4",  and  other  dimensions  equally  ample. 

Sections  of  rail  are  of  infinite  variety  and  weights,  adapted  to  the  class  of 
railroads  on  which  they  are  to  be  used,  and  the  loads  and  speed  of  trains  to 
which  they  are  to  be  subjected.  For  roads  of  the  common  gauge,  the  weight 
of  rails  is  from  56  to  70  Ibs.  per  yard.  The  joints  are  made  with  a  fish-plate. 

Figs.  883,  884,  and  885  are  the  elevation,  section,  and  plan  of  the  standard 
rail-joint  of  the  New  York,  West  Shore  and  Buffalo  Railroad. 


ENGINEERING  DRAWING. 


407 


FIG.  883. 


FIG.  885. 


ROOFS   AND   BRIDGES. 

At  pages  238  and  239  will  be  found  illustrations  of  the  trussing  of  wooden 
beams.  These  are  simple  forms,  which  may  be  used  in  roofs  or  bridges,  and 
rules  are  given  for  the  proportion  of  parts.  Soiled  I-beams  or  plate-girders 
will  serve  also  for  floor-beams  and  moderate  spans,  but  with  modern  necessities 
much  more  complicated  structures  are  required. 

On  the  General  Principles  of  Bracing. — Let  Fig.  886  be  the  elevation  of  a 
common  roof-truss,  and  let  a  weight,  W,  be  placed  at  the  foot  of  one  of  the  sus- 
pension-rods. Now,  if  the  construction  consisted  merely  of  the  rafter  C'B, 
and  the  collar-beam  C'  C,  resting  against  some  fixed  point,  then  the  point  B 
would  support  the  whole  downward  pressure  of  the  weight ;  but  in  consequence 
of  the  connection  of  the  parts  of  the  frame,  the  pressure  must  be  resolved  into 
components  in  the  direction  0'  A  and  C'  B  ;  C'  b  will  represent  the  pressure  in 

the  direction  C'  B,   C'  w  the 
portion    of   the   weight   sup- 


FIG.  886. 


FIG.  887. 


ported  at  B,  C'  a  the  pressure  in  the  direction  C'  A,  and  w  W  the  portion  of 
the  weight  supported  on  A.  The  same  resolution  obtains  to  determine  the 
direction  and  amount  of  force  exerted  on  a  bridge-truss  of  any  number  of 


408 


ENGINEERING  DRAWING. 


panels,  by  a  weight  placed  at  any  pointy  of  its  length  (Fig.  887).  In  either 
case,  the  effect  of  the  oblique  form  0'  A  upon  the  angle  C  is  evidently  to  force 
it  upward  ;  that  is,  a  weight  placed  at  one  side  of  the  frame  has,  as  in  case  of 
the  arch,  a  tendency  to  raise  the  other  side.  The  effect  of  this  upward  force  is 
a  tension  on  a  portion  of  the  braces,  according  to  the  position  of  the  weight ; 
but  as  braces,  from  the  manner  in  which  they  are  usually  connected  with  the 
frame,  are  not  capable  of  opposing  any  force  of  extension,  it  follows  that  the 
only  resistance  is  that  which  is  due  to  the  weight  of  a  part  of  the  structure. 


FIG.  890. 


FIG.  889. 


Figs.  888  and  889  illustrate  the  effects  of  overloading  at  single  points  such 
forms  of  construction.  Such  an  unequal  loading  on  trusses  requires  that  a 
portion  of  the  load  W  be  tranferred  to  each  point  of  support  inversely  propor- 
tionate to  the  distances  of  the  weight  from  each 
support.  The  above  trusses  are  not  prepared  to 
transfer  this  weight  to  but  one  support.  To  rem- 
edy the  difficulty,  it  will  be  necessary  to  add  braces 
running  in  the  opposite  direction,  as  shown  by 
dotted  lines  (Fig.  890),  at  every  point  subject  to  the  above  distortion.  These 
are  called  counter-braces. 

To  prevent  the  braces  from  becoming  loose  when  the  counter-braces  are  in 
action,  it  is  always  customary  to  give  the  braces  and  counter-braces  an  initial 
compression,  by  putting  a  moderate  tension  on  the  suspension-rods.  In  this  case, 
therefore,  the  passage  of  a  load  would  produce  no  additional  strain  upon  any  of 
the  timbers,  but  would  tend  to  relieve  the  counters.  The  counter-braces  do 
not,  of  course,  assist  in  sustaining  the  weight  of  the  structure  ;  on  the  contrary, 
the  greater  the  weight  of  the  structure  itself,  the  more  will  the  counter-braces 
be  relieved. 

If,  instead  of      £_ 4 ^ 2 /  2.'      _._£ 4-' £ 

the  counter- 
braces,  the  braces 
themselves  are 
made  to  act  both 
as  ties  and  struts, 
as  has  been 
done  sometimes 
in  iron  bridges 
and  trusses,  then 
the  upward  force  will  be  counteracted  by  the  tension  of  the  brace. 

Suppose  a  system  to  be  composed  of  a  series  of  suspension-trusses,  as  in  Fig. 
891,  in  which  the  load  is  uniformly  distributed.  If  we  represent  the  load  at 


FIG.  891. 


ENGINEERING  DRAWING.  409 

each  of  the  points,  4,  3,  2,  1,  2',  etc.,  by  1,  the  load  at  4  will  be  supported  ^  upon 
a  and  %  upon  3  ;  hence  the  strut  3  will  have  to  support  a  load  of  1  -f-  "5  =  1  *5  ; 
of  this,  f  will  be  supported  by  2  and  -J-  by  a  ;  f  of  1*5  =  1,  1  +  1  =  2,  load  on 
strut  2  ;  f  of  this  load,  or  1  *5,  will  be  supported  at  1,  and  since  from  the  op- 
posite side  there  is  an  equal  force  exerted  at  1,  therefore  the  strut  1  supports 
1+1-5  +  1-5  =  4. 

The  tension  on  the  rod  c-2  =  2 

cl 

2-3  =  2%  " 
"  "  3-4  =  3     " 


If  this  construction  be  reversed,  the  parts  which  now  act  as  ties  must  be  made 
as  braces,  and  braces,  ties  ;  then  we  have  a  roof-truss,  and  the  force  exerted  on 
the  several  parts  may  be  estimated  in  a  similar  way  as  for  the  suspension-truss. 

It  is  evident  that  neither  of  these  constructions  would  serve  for  a  bridge- 
truss,  subject  to  the  passage  of  heavy  loads,  but  is  only  fit  to  support  uniform 
and  equally  distributed  loads. 

To  frame  a  construction  so  that  it  maybe  completely  braced  —  that  is,  under 

the  action  of  any  arrangement  of  forces  —  the  angles  must  not  admit  of  altera- 

tion, and  consequently  the  shape  can  not.     The  form  should  be  resolvable  into 

either  of  the  following  elements  (Figs.  892,  893,  and  894)  : 


FIG.  892.  FIG.  893.  FIG.  894. 

In  these  figures,  lines  -  -  represent  parts  required  to  resist  com- 
pression ;  lines  parts  to  resist  tension  only ;  lines  •  parts 

to  resist  both  tension  and  compression. 

It  is  evident  that,  in  a  triangle  (Fig.  892),  an  angle  can  not  increase  or 
diminish,  without  the  opposite  angles  also  increasing  or  diminishing.  In  the 
form  Fig.  893,  a  diagonal  must  diminish  ;  in  Fig.  894  a  diagonal  must  extend, 
in  order  that  any  change  of  form  may  take  place.  Consequently,  all  these 
forms  are  completely  braced,  as  each  does  not  permit  of  an  effect  taking  place, 
which  would  necessarily  result  from  a  change  of  figure.  Hence,  also,  any  sys- 
tem composed  of  these  forms,  properly  connected,  breaking  joint  as  it  were 
into  each  other,  must  be  braced  to  resist  -the  action  of  forces  in  any  direction  ; 
but  as  in  general  all  bridge-trusses  are  formed  merely  to  resist  a  downward 
pressure,  the  action  on  the  top  chord  being  always  compression,  it  is  not  neces- 
sary that  these  chords  should  act  in  both  capacities. 

Roofs. — The  roofs  of  city  dwellings  and  stores  are  generally  flat,  that  is, 
with  but  very  little  inclination,  from  half  an  inch  to  two  inches  per  foot, 
merely  sufficient  to  discharge  the  water.  The  beams  are  laid  from  wall  to  wall, 
the  same  as  floor-timbers,  but  usually  of  less  depth,  or  at  greater  distances  be- 
tween centers,  and  with  one  or  two  rows  of  bridging. 

Figs.  895,  896,  and  897  represent  side  or  portions  of  side  elevations  of  the 
usual  form  of  framed  roofs.  The  same  letters  refer  to  the  same  parts  in  all 


±10 


ENGINEERING  DRAWING. 


the  figures.  T  T  are  the  tie-beams,  R  R  the  main  rafters,  rr  the  jack-rafters, 
PP  the  plates,  pp  the  purlines,  K  K  the  Icing-posts,  kk  Icing-bolts,  q  q  queen- 
bolts— both  are  called  suspension- bolts — C  C  the  collar  or  straining  beams,  B  B 
braces  or  struts,  b  b  ridge-boards,  e  corbels. 


FIG.  895. 


The  pitch  of  the  roof  is  the  inclination  of  the  rafters,  and  is  usually  desig- 
nated in  reference  to  the  span,  as  £,  ^-,  f,  etc.,  pitch  ;  that  is,  the  height  of  the 
ridge  above  the  plate  is  ^,  i,  f ,  etc.,  of  the  span  of  the  roof  at  the  level  of  the 
plate.  The  steeper  the  pitch  of  the  roof,  the  less  the  thrust  against  the  side- 
walls,  the  less  likely  the  snow  or  water  to  lodge,  and  consequently  the  tighter 
the  roof.  For  roofs  covered  with  shingles  or  slate,  in  this  portion  of  the  coun- 


Fia.  896. 


try,  it  is  not  advisable  to  use  less  than  J  pitch  ;  above  that,  the  pitch  should  be 
adapted  to  the  style  of  architecture  adopted.  The  pitch  in  most  common  use 
is  £  the  span. 

Fig.  895  represents  the  simplest  framed  roof  :  it  consists  of  rafters,  resting 
upon  a  plate  framed  into  the  ceiling-beam  ;  this  beam  is  supported  by  a  sus- 
pension-rod, k,  from  the  ridge,  but,  if  supported  from  below,  this  rod  may  be 
omitted.  As  shown,  the  rafters  are  to  be  spaced  from  1  to  2  feet  centers,  and 
the  tie-beams  at  intervals  of  from  6  to  8  feet  :  the  roof  cover  to  be  of  boards 


ENGINEERING  DRAWING. 


411 


nailed  directly  to  the  rafters.  This  form  of  construction  is  sufficient  for  any 
roof  of  less  than  25  feet  span,  and  of  the  usual  pitch,  and  may  be  used  for  a  40- 
foot  span  by  increasing  the  depth  of  the  rafters  ;  deep  rafters  should  always  be 
bridged.  By  the  introduction  of  a  purline  extending  beneath  the  center  of  the 


FIG.  897 


rafter^  supported  by  a  brace  to  the  foot  of  the  suspension-rod,  as  shown  in  dot- 
ted line,  the  depth  of  the  rafters  may  obviously  be  reduced.  It  often  happens 
that  the  king-bolt  may  interfere  with  the  occupancy  of  the  attic  ;  in  that  case 
the  beam  is  otherwise  supported.  Again,  it  may  be  necessary  that  the  tie- 
beam,  which  is  also  a  ceiling  and  floor  beam,  should  be  below  the  plate  some  2 
to  4  feet ;  in  that  case,  the  thrust  of  the  roof  is  resisted  (Fig.  898)  by  bolts, 
b  b9  passing  through  the  plate  and  the  beam,  and  by  a 
collar-plank,  C,  spiked  on  the  sides  of  the  rafters,  high 
enough  above  the  beam  to  afford  good  head-room.  For 
roofs  f  pitch  and  under  20  feet  span,  the  bolts  are  un- 
necessary, the  collar  alone  being  sufficient. 

Fig.  896  represents  a  roof,  a  larger  span  than  Fig. 
895  ;  the  frame  may  be  made  very  strong  and  safe  for 
roofs  of  60  feet  span.  King-bolts  or  suspension-rods 

are  now  oftener  used  than  posts,  with  a  small  triangular  block  of  hard  wood 
or  iron,  at  the  foot  of  the  bolts,  for  the  support  of  the  braces.  The  objection 
to  this  form  of  roof  is  that  the  framing  occupies  all  the  space  in  the  attic  ; 
on  this  account  the  form,  Fig.  897,  is  preferred  for  roofs  of  the  same  span, 
and  is  also  applicable  to  roofs  of  at  least  75  feet  span,  by  the  addition  of  a 
brace  to  the  rafter  from  the  foot  of  the  queen-bolt.  The  collar-beam  (Fig. 
900)  is  also  trussed  by  the  framing  similar  to  Fig.  896. 

In  many  church  and  barn  roofs  the  tie-beam  is  cut  off  (Fig.  899)  ;  the 
queen-post  being  supported  on  a  post,  or  itself  extending  to  the  base,  with  a 
short  tie-rod  framed  into  it  from  the  plate. 


412 


ENGINEERING  DRAWING. 


Figs.  901  and  902  are  representations  of  the  feet  of  rafters  on  an  enlarged 
scale.     In  Fig.  901,  the  end  of  the  rafter  does  not  project  beyond  the  face  of 


the  plate ;    the   cove  is  formed   by  a  Fl°-  90° 

small  triangular,  or  any  desirable  form 

of  plank,  framed  into  the  plate.     The  form  given  to  the  foot  of  the  rafter  is 

called  a  crow-foot.     In  Fig.  902,  the  rafter  itself  projects  beyond  the  plate  to 

form  the  coving.     Fig.  903  represents  a  front  and  side  elevation  and  plan  of 

the  foot  of  a  main  rafter,  showing  the  form 

of  tenon,  in  this  case  double ;  a  bolt,  passing 


nrn 


FIG.  901. 


FIG.  902. 


FIG.  903. 


through  the  rafter  and  beam,  retains  the  foot  of  the  former  in  its  place.     Fig. 
904  represents  the  foot  of  a  main  rafter,  with  a  wooden  shoe  too  short  at  #, 

outside  of  the  rafter ;   it  should  be  framed  as  in  Fig. 

903.     In  Fig.  901,  of  a  similar  construction  to  Fig.  895, 

the  tie-rod  passes  directly  through  the  plate.     In  general, 

when  neither  ceiling  nor  flooring  is 

supported  by  the  tie-beam,  a  rod  is 

preferable. 

FIG.  904. 


0 

n 

L° 

FIG.  905. 


FIG.  906. 


FIG.  907. 


Roofs  are  now  very  neatly  and  strongly  framed  by  the  introduction  of  cast- 
iron  shoes  and  abutting  plates  for  the  ends  of  the  braces  and  rafters.     Fig.  905 


ENGINEERING  DRAWING. 


413 


represents  the  elevation  and  plan  of  a  cast-iron  king-head  for  a  roof  similar  to 
Fig.  896  ;  Fig.  906,  that  of  the  brace-shoe  ;  Fig.  907,  that  of  the  rafter-shoe 


FIG.  908. 


FIG.  909. 


FIG.  910. 


for  the  same  roof ;  Fig.  908,  the  front  and  -side  elevation  of  the  queen-head 
of  roof  similar  to  Fig.  897 ;  and  Fig.  909,  elevation  and  plan  of  queen  brace- 
shoe. 

Fig.  910  represents  the  section  of  a  rafter-shoe  for  a  tie-rod  ;  the  side 
flanches  are  shown  in  dotted  line. 

On  the  size  and  the  proportions  of  the  different  members  of  a  roof  :  Tie- 
beams  are  usually  intended  for  a  double  purpose,  and  are  therefore  affected  by 
two  strains  :  one  in  the  direction  of  their  length,  from 
the  thrust  of  the  rafters  ;  the  other  a  cross-strain,  from 
the  weight  of  the  floor  and  ceiling.     In  estimating  the 
size  necessary  for  the  beam  the  thrust  need  not  be 
considered,  because  it  is  always  abundantly  strong  to 
resist  this  strain,  and  the  dimensions  are  to  be  deter- 
mined as  for  a  floor-beam  merely,  each  point  of  sus- 
pension being  a  support.     When  tie-rods  are  used,  the 

strain  is  in  the  direction  of  their  length  only,  and  their  dimensions  can  be 
calculated,  knowing  the  pitch,  span,  and  weight  of  the  roof  per  square  foot, 
and  the  distance  apart  of  the  ties,  or  the  amount  of  surface  retained  by 
each  tie. 

The  weight  of  the  wood-work  of  the  roof  may  be  estimated  at  40  pounds 
per  cubic  foot  ;  slate  at  7  to  9  pounds,  shingles  at  1J  to  2  pounds  per  square 
foot.  The  force  of  the  wind  may  be  assumed  at  15  pounds  per  square  foot. 
The  excess  of  strength  in  the  timbers  of  the  roof,  as  allowed  in  all  calculations, 
will  be  sufficient  for  any  accidental  and  transient  force  beyond  this.  Knowing 
the  weights,  pressures,  and  their  directions  on  parts  of  a  roof,  their  stresses  may 
be  determined  by  the  parallelograms  of  forces  and  dimensions  proportioned  to 
the  strength  of  the  materials  of  which  the  roof  is  composed.  It  will  generally 
be  sufficient  for  the  draughtsman  to  have  practical  examples  of  construction  to 
draw  from.  Dimensions  are  therefore  given  of  the  parts  of  wooden  roofs  already 
illustrated.  With  further  examples  of  actual  constructions,  the  beams  are  usu- 
ally proportioned  to  the  weight  that  they  are  to  sustain  in  floors  and  load,  but 
where  tie-rods  are  used,  the  stress  upon  them  may  be  determined  by  the  follow- 
ing rule  : 

Rule. — Multiply  one  half  the  weight  of  the  roof  and  load  by  one  half  the 
span,  and  divide  the  product  by  the  rise  or  height  of  ridge  above  eaves. 

Gwilt,  in  his  "Architecture,"  recommends  the  following  dimensions  for 
portions  of  a  roof  : 


414: 


ENGINEERING  DRAWING, 


Span. 

Form  of  Koof. 

Kafters. 

Braces. 

Posts. 

Collar-beams. 

Feet. 

Inches. 

Inches. 

Inches. 

Inches. 

25 

Fig.  896, 

5x4 

5x3 

5x5 

30 

u 

6x4 

6x3 

6x6 

35 

Fig.  897, 

5x4 

4x2 

4x4 

7x4 

45 

u 

6x5 

5x3 

6x6 

7x6 

50 

2  sets  of  queen-posts, 

8x6 

5x3 

~>         O               AC 

9x6 

{     8  x  4     f 

60 

u                          a 

8x8 

6x3 

j  10  x   8      j 
"j  10  x  4      f 

11   x  6 

These  dimensions,  for  rafters,  are  somewhat  less  than  the  usual  practice  in 
this  country  ;  no  calculations  seem  to  have  been  made  for  using  the  attic.  An 
average  of  common  roofs  here  would  give  the  following  dimensions  nearly  :  30 
feet  span,  8X5  inches  ;  40  feet,  9  X  6  ;  50  feet,  10  X  7  ;  60  feet,  11  X  8  ; 
collar-beams  the  same  size  as  main  rafters.  Roof -frames  from  8  to  12  feet  from 
center  to  center. 

Dimensions  for  jack-rafters,  15  to  18  inches  apart : 


For  a  bearing  of  12  feet. ...  6x3  inches. 
"  "  10    "  .      .  9  x  3      " 


For  a  bearing  of    8  feet. ...  4x3  inches. 
"  "  20    "  .    ..10  x  3      " 


Purlines  : 

Length  of  Bearing. 

Distances  apart  in  Feet. 

Feet. 

6 

8 

10 

12 

8 

7x5 

8x5 

9x5 

9x6 

10 

9x5 

10  x  5 

10  x  6 

11   x   6 

12 

10   X  6 

11   x  6 

12  x  7 

13  x   8 

The  pressure  on  the  plates  is  transverse  from  the  thrust  of  the  rafters,  but 
in  all  forms  except  Fig.  895,  owing  to  the  notching  of  the  rafters  on  the  pur- 
lines,  this  pressure  is  inconsiderable.  The  usual  size  of  plates  for  Figs.  895  and 
896  is  6  x  6  inches. 

In  the  framing  of  roofs,  it  is  now  customary,  for  roofs  of  mills,  to  omit 
purlines,  jack-rafters,  and  plates,  and  make  the  roof-boards  of  plank  stiff 
enough  to  supply  their  places,  from  2"  to  3"  thick  (according  to  the  space 
between  the  frames),  tongued  and  grooved,  and  strongly  spiked  to  the  main 
rafters. 

The  dimensions  of  rafters  depend  on  the  distances  between  their  supports 
and  between  centers.  The  depth  in  all  such  cases  to  be  greater  than  the  width  ; 
2  to  6  inches  may  be  taken  as  the  width,  8  to  12  for  the  depth. 

When  there  are  no  purlines,  and  the  roof  is  covered  with  plank,  there  is  no 
need  of  plates  ;  the  plank  forms  a  deep  beam,  and,  if  the  ends  of  the  frame  are 
secured,  there  may  be  no  need  of  intermediate  ties. 

Iron  Roofs. — Roofs  of  less  than  30  feet  span  are  often  made  of  corrugated 
iron  alone,  curved  into  a  suitable  arc,  and  tied  by  bolts  passing  through  the 
iron  about  2  to  4  feet  above  the  eaves. 


ENGINEERING  DRAWING. 


415 


Fig.  911  represents  the  half  elevation  of  an  iron  roof  of  a  forge  at  Paris  ; 
Figs.  912,  913,  and  914,  details  on  a  larger  scale.     This  is  a  common  type  of 


iron  roof,  consisting  of  main  rafters,  E,  of  the  I-section  (Fig.  914), 
trussed  by  a  suspension-rod,  and  tied  by  another  rod.     The  purlines 
are  also  of  I-iron,  secured  to  the  rafters  by  pieces  of  angle-iron  on  each 
side ;   and  the  roof  is  cov- 
ered with  either  sheet-iron  resting  ^4 
on  jack-rafters,  or  corrugated  iron 
extending  from  purlin  e  to  purline. 
The  rafter-shoe,  A,  and  the  strut, 


S,  are  of  cast-iron ;  all  the  other  portions 
of  the  roof  are  of  wrought-iron.    In  Amer- 
ican practice  it  is  usual 

to  make  the  strut  of  FIG.  913. 

wrought-iron,  with   a 

single  pin  connection  at  its  foot,  instead  of  as  in  the  figure. 
The  surface  covered  by  this  particular  roof  is  53  metres 
(164  feet)  long  and  30  metres  (98J-  feet)  wide.  There  are 
eleven  frames,  including  the  two  at  the  ends,  which  form 
the  gables. 

The  following  are  the  details  of  the  dimensions  and 
FIG.  914.  weights  of  the  different  parts  : 


416 


ENGINEERING  DRAWING. 


Pounds. 

2  rafters,  0'72  feet  deep  ;  length  together,  99*1  feet 1,751 

5  rods,  0-13  feet  diameter ;  length  together,  131'4  feet 882 

16  bolts,  0-13  feet  diameter 79 

8  bridle-straps,  0-24  x  -05 123 

2  pieces,  0'46  thick,  connecting  the  rafters  at  the  ridge,  >  8g 

4  pieces,  0'46  thick,  at  the  foot  of  the  strut j 

4  pieces,  0*36  thick,  uniting  the  rafters  at  the  junction  in  the  strut — together  with 

their  bolts  and  nuts 176 

2  cast-iron  struts 308 

2  rafter-shoes 287 

Total  of  one  frame 3,694 

16  purlines,  1  ridge-iron,  each  0'46  deep,  17'2  long 2,985 

Bolts  for  the  same 64 

16  jack-rafters,  I-iron,  0*16  deep 2,489 

Weight  of  iron  covering,  including  laps,  per  square  foot 2-88 

Koofs  are  sometimes  made  with  deep  corrugated  main  rafters  with  flat  iron 
between,  or  purlines  and  corrugated  iron  for  the  covering.  The  great  objec- 
tion to  iron  roofs  lies  in  the  condensation  of  the  interior  air  by  the  outer  cold, 
or,  as  it  is  termed,  sweating  ;  on  this  account,  they  are  seldom  used  for  other 

_     buildings  than  boil- 
**^     er-houses  or  depots, 
except  a  ceiling  be 
made  below  to  pre- 
vent the  contact  of 
the  air  inside  with 
-<^     the  iron. 
=_,          Fig.    915   is   an 
FIG.  915.  elevation  of  one  of 

the  three  panels  of 

one  of  the  cast-iron  girders  for  connecting  the  columns,  and  carrying  the  trans- 
verse main  gutters,  which  supported  the  roof  of  the  English  Crystal  Palace. 

Figs.  916  to  921  are  sections  of  va- 
1*  rious  parts  on  an  enlarged  scale. 

The  depth  of  the  girder  was  3 
feet,  and  its  length  was  23  feet 
3f  inches.  The  sectional  area  of 
the  bottom  rail  and  flange  in  the 
center  (Fig.  917)  was  6^  square 


FIG.  919. 


FIG.  920. 


FIG.  921. 


inches ;  the  width  of  both  bottom  and  top  rail  (Fig.  916)  was  reduced  to  3 
inches  at  their  extremities. 


ENGINEERING  DRAWING. 


417 


27 


4:18 


ENGINEERING  DRAWING. 


The  weight  of  these  girders  was  about  1,000  pounds,  and  they  were  proved 
by  a  pressure  of  9  tons,  distributed  on  the  center  panel. 

A  second  series  of  girders  were  made  of  similar  form,  but  of  increased 
dimensions  in  the  section  of  their  parts.  Their  weight  averaged  about  1,350 
pounds,  and  were  proved  to  15  tons.  A  third  series  weighed  about  2,000 
pounds,  and  were  proved  to  22%  tons. 

Figs.  922  to  927  are  the  elevation  and  details  for  an  iron  roof-truss,  for 


FIG.  929. 


ENGINEERING  DRAWING.  419 

wood,  slate,  or  corrugated  iron  covers,  built  by  the  Missouri  Valley  Bridge  and 
Iron  Works,  A.  S.  Tulloch,  engineer. 

Figs.  928  and  929  are  sections  and  details  of  the  trusses  for  sustaining 
the  roof  and  floor  of  the  new  English  High  and  Latin  School  Gymnasium, 
Boston,  Massachusetts.  The  object  of  sustaining  the  gymnasium-floor  by  rods 
was  to  secure  a  drill-hall  for  the  military  exercises  of  the  school,  and  trusses 
were  designed  to  have  sufficient  strength  to  resist  the  vibration  of  the  floor. 
As  the  trusses  were  to  be  in  sight,  a  central  column  of  cast-iron  was  introduced 
to  sustain  the  center  of  the  top  chord,  instead  of  some  wrought-iron  construc- 
tion less  pleasing  to  the  eye,  with  lattice  between  the  main  diagonals  to  enable 
them  to  act  as  counters,  instead  of  a  more  complicated  construction  introducing 
counters,  and  a  3-J-inch  gas-pipe  for  horizontal  bracing-struts.  The  floor-sus- 
taining rods  all  have  upset  ends,  and  at  their  tops  pass  through  ornamental 
foliated  castings,  but  their  connection  with  the  trusses  is  wholly  of  wrought-iron. 

The  top  chords  consist  of  two  nine-inch  channel-irons  weighing  50  pounds 
per  yard,  and  one  plate  12  X  f  inches.  The  end-posts  have  the  same  section. 
The  bottom  chord  consists  of  four  bars  2-J-  X  1  inch  at  the  shallow  end  of  the 
truss,  and  four  bars  2^  X  f  of  an  inch  at  the  deep  end  of  the  truss.  The  diago- 
nals are  two  bars  3X1  inch  at  deep  end  of  truss,  and  two  bars  3  X  i  inch  at 
shallow  end  of  truss.  The  pins  are  all  2-J  inches  diameter. 

These  trusses  were  designed  and  constructed  by  D.  H.  Andrews,  C.  E.,  of 
the  Boston  Bridge  Works. 

In  order  to  secure  free  space  in  the  room  beneath  the  roof,  it  is  my  practice 
to  construct  a  roof  or  bridge  truss  above,  and  suspend  from  it  the  roof  framed 
as  a  floor,  with  such  pitch  as  is  requisite  to  shed  rainfall.  In  this  form  of 
construction  the  span  of  the  unobstructed  space  required  is  readily  met  by  the 
truss  construction. 

Fig.  930  is  a  half  cross-section  of  a  two-story  freight-shed  for  the  New  York, 
Lake  Erie  and  Western  Eailroad.  It  is  a  simple  and  cheap  construction  of 
wood,  readily  framed  and  put  together.  The  shed  rests  upon  a  pile-dock. 
The  platform  for  the  reception  of  freight  is  4  feet  above  the  dock-planking, 
and  about  26  feet  wide,  with  occasional  inclined  runs  for  the  transfer  of  freight 
to  or  from  vessels. 

Bridge-Trusses. — Whatever  maybe  the  form  of  truss  or  arrangement  of 
the  framing,  provided  its  weight  is  supported  only  at  the  ends,  the  tension 
of  the  lower  chord,  or  the  compression  of  the  upper  chord  at  center,  may  be 
determined  by  this  common  rule  : 

Rule. — The  sum  of  the  total  weight  of  the  truss,  and  the  maximum  dis- 
tributed load  which  it  will  be  called  on  to  bear,  multiplied  by  the  length  of  the 
span,  and  divided  by  8  times  the  depth  of  the  truss  in  the  middle,  the  quotient 
will  be  the  tension  of  lower  chord  and  compression  of  upper  at  the  middle.  In 
nearly  all  the  forms  of  diagonal  bracing,  if  the  uniform  load  be  considered  as 
acting  from  the  center  toward  each  abutment,  each  tie  or  brace  sustains  the 
whole  weight  between  it  and  the  center,  and  the  strain  is  this  weight  multiplied 
by  the  length  of  tie  or  brace,  divided  by  its  height.  Any  diagonals,  equally 
distant  from  the  center,  sustain  all  the  intermediate  load  :  if  rods,  as  in  Fig. 
932,  by  tension  ;  if  braces,  as  in  Fig.  931,  by  compression. 


ENGINEERING  DRAWING. 


CROSS-SECTION  OF  ONE   HALF  OF  A  FREIGHT-SHED,  NEW  YORK,   LAKE  ERIE 
AND  WESTERN  RAILROAD. 


II    U    U 


FIG.  930. 


ENGINEERING  DRAWING. 


421 


It  follows,  therefore,  that  in  all  these  trusses  the  upper  and  lower  chords 
should  be  stronger  at  the  center  than  at  the  ends,  while  diagonals  should  be 
largest  at  the  abutments.  Unless  the  weight  of  the  bridge  is  great  compared 
with  the  moving  loads,  counter-braces  become  necessary. 

The  general  rule  adopted  in  the  construction  of  the  Howe  truss  is,  to  make 
the  height  of  the  truss  -J  of  the  length  up  to  60  feet  span  ;  above  this  span  the 
trusses  are  21  feet  high,  to  admit  of  a  system  of  lateral  bracing,  with  plenty  of 
head  clearance  for  a  person  standing  on  the  top  of  a  freight-car.  From  175 
feet  to  250  feet  span,  height  of  truss  gradually  increased  up  to  25  feet.  Moving 
load  for  railroad -bridge  calculated  at  1  ton  per  running  foot.  Center  to  center 
of  panels  not  exceeding  11  feet. 

Wooden  Truss-Bridges. — Fig.  931  is  the  elevation  of  a  few  panels  of  a  Howe 
truss,  and  Fig.  932  of  a  Pratt  truss.  The  Howe  truss  is  by  far  the  most  popu- 


FIG.  931. 


FIG.  932. 


lar  of  all  wooden  trusses,  being  readily  framed  and  put  together,  uniting  great 
strength  with  simplicity  of  construction. 

Fig.  933  is  the  side  elevation  of  three  of  the  five  panels  of  a  Howe  truss 
highway-bridge  of  the  New  York,  Lake  Erie  and  Western  Railroad.  Fig.  934 
is  a  cross-section.  It  will  be  observed  that  there  is  a  section  of  3"  plank  laid 
close,  and  another  beneath,  laid  with  spaces ;  these  planks  are  laid  diagonally 
across  the  floor-beams,  and  at  right  angles  to  each  other,  and  are  made  to  act 
as  lateral  bracing.  Fig.  935  are  the  details  of  the  abutment  end  of  bridge ; 
the  foot  of  the  brace  rests  on  a  cast-iron  shoe.  The  length  over  all — that  is, 
including  the  portions  on  the  abutment — is  81'  2ff,  or  75  feet  between  abut- 
ments, usually  designated  as  the  span. 

Figs.  936,  937,  and  938  are  the  side  elevation,  floor  cross-section,  and  plan 
of  floor  and  bottom  chord  of  three  of  the  twelve  panels  of  a  single-track  rail- 
way Howe  truss.  Their  length  is  each  10'  10^".  The  center  braces  are  two, 
7"  X  10" ;  the  center  rods  three,  1-J-*  diameter.  The  counters,  each  one  6"  X 
8" ;  lateral  brace  top  and 
bottom,  6"  X  6" ;  rods  1£ 
inch ;  top  chord,  four  pieces, 
7"  X  12";  bottom  chord, 
four  pieces,  1"  X  15" ;  floor- 
beams,  1"  X  16".  The  shoes, 
splices,  and  blocks  between 
chord-timber  are  of  cast-iron. 

In  the  earlier  practice  the  angle-blocks  were  of  oak,  and  the  splices  made  as 
in  Fig.  939.  Both  of  these  were  satisfactory. 


FIG.  939. 


422 


ENGINEERING  DRAWING. 


ENGINEERING  DRAWING. 


423 


TXg 


fl 


=i 


F 


A 


JZ 


uU'll   i  i        !,,i     » 


.J9f  •*«•«*•  «<! 

FIG.  938. 


424 


ENGINEERING  DRAWING. 


Combination  Truss. — Figs.  940  and  941  are  the  elevation  and  plan,  and 
Figs.  942  and  943  the  details  of  the  combination  or  composite  truss,  which 
owes  its  name  to  the  use  of  the  two  materials,  wood  and  wrought-iron,  in 


somewhat  near  the  same  proportion  in  its 
construction,  the  tension  members  being  of 
iron  and  the  compression  of  wood.  The  cen- 
tral braces,  which  are  subjected  alternately 
to  tensile  and  compression  stresses,  may  be  of 
wood  with  iron  rods,  or  wrought-iron  only. 
This  class  is  entirely  American  in  practice, 
and  embodies,  as  will  be  seen  in  the  details, 
an  essentially  American  feature,  of  pin  con- 
nections. The  bridge  illustrated  is  in  30-foot 
panels,  six  to  the  full  length.  The  shoes  and 
splices  are  of  cast-iron. 

Iron  Bridges. — When  the  span  is  of  mod- 
erate extent,  the  load  can  be  safely  carried  by 
beams  put  together  at  the  works  and  trans- 
ferred to  the  road  in  complete  form.  Web  or 
lattice  girders  are  used,  put  together  with 
rivets. 

FIG  942  Figs.  944,  945,  and  946  are  the  outside 

elevation,  plan  of  top  bracing,  and  plan  of 

bottom  bracing  of  one  half  a  deck  plate-girder  railway-bridge,  42'  6"  over  all, 
or  40  feet  span  or  effective  length.     Figs.  947  and  948  are  the  end-elevation 


ENGINEERING  DRAWING. 


425 


FIG.  943. 


FIG.  945. 


ENGINEERING   DRAWING. 


FIG.  946. 


FIG.  947. 


FIG.  948. 


and  a  section  near  the  center.     This  and  the  following  illustrations  are  taken 
from  "  The  American  Engineer/'  and  the  bill  of  material  given  is  as  follows  : 

BILL   OF   MATERIAL  FOR  DECK   GIRDER,   42'  6"  LONG  OVER  ALL. 


No. 

IHMKXCTOVS. 

Weight. 

For  what  used. 

Pounds. 

4 

Bars,  angle,  4"  x  5"—  14'2  Ibs.  x  14'  0" 

Top  flanges. 

4 

"        "           "                 "         x  28'  0" 

2,386 

('                  U 

4 

"        "     4"  x  6"—  24-£lbs.  x  14'  0" 

Bottom  flanges. 

4 

"         '           "                "         x  23'  0" 

4,116 

u              u 

4 

"         '     3-J-"  x  5"—  20-8  Ibs.  x     2'  8" 

222 

Angle-covers. 

4 

"         '     3£"x4"—  12     Ibs.  x     2'  8" 

128 

a              a 

32 

«         '     3"    X4"—  8-3  Ibs.  x3  10J" 

1,029 

Ends  and  stiffeners. 

16 

"  '      '     3"    x3"—  7'2  Ibs.  x7'    5" 

Lateral. 

2 

"         '            "              "          x5'    4" 

931 

Center-bracing. 

2 

"         <     2-fc"  x  2V—  5  Ibs.   x  5'  9" 

"           " 

4 

End-bracing. 

4 

303 

"         '' 

4 

Plates,  48"  x  f"  x  21'  0" 

5,040 

Webs. 

2 

14"  x  |"  x  29'  0" 

1,691 

Top  flanges. 

4 

12"  x  1"   x     3'  4" 

200 

Joint-covers. 

4 

10"  x  ,56  '  x     6'  5" 

267 

End-bracing. 

7 

9"  x  4"  x     1'  9" 

Lugs. 

1 

"         x      2'  0" 

c* 

1 

"         x     1'  0" 

172 

i. 

2 

8"  x  |"  x     1'  0" 

20 

" 

4 

14"  x  i"  x     2'  0" 

187 

Bearing-plates. 

32 

Flat  bars,  3"  x  f"  x  3'  4" 

667 

Fillers. 

24 

"         3"  x  |"  x  3'  4" 

400 

Inside  stiffeners. 

4 

"         6"  x  I"  x  2'  5" 

97 

Joints. 

17,856 

Rivets  6  per  cent                

1,070 

18,926 

Cast  bearing-blocks   @  200 

800 

Total  weight   ...          

19,726 

ENGINEERING  DRAWING. 

OUTSIDE  ELEVATION. 


4:27 


FIG.  949. 
PLAN  OF  TOP  BRACING. 


FIG.  950. 
PLAN  OF  BOTTOM  BRACING. 


CROSS-SECTION  NEAR  CENTER 


FIG.  952. 

Figs.  949,  950,  and  951  are  the  out- 
side elevation,  plan  of  top  and  bottom 
bracing  of  one  half  a  deck  lattice-girder 
railway-bridge,  the  same  span  as  Fig. 
944  above,  and  intended  to  carry  the 
-same  load — rolling  4,000  pounds,  and 


FIG.  953. 
JOINT  %  RIVETS. 


o  o    o 
o  o    o     o 
o    o     o  o    o    o    o" 


00    o    o    o  o 


o    o 


FIG.  954. 


4:28 


ENGINEERING  DRAWING. 


dead  load  900  pounds  per  lineal  foot.     Figs.  952  and  953  are  the  end  elevation 
and  cross-section  near  center,  and  Fig.  954  one  of  the  joints. 

BILL  OF  MATERIAL  FOR  DECK  LATTICE-BRIDGE,  42'  6"  LONG  OVER  ALL. 


No. 

DIMENSIONS. 

Weight. 

For  what  used. 

Pounds. 

4 

Plates,  12"  x 

i" 

X 

13'  6" 

Chords  —  webs. 

4 

M 

X 

28'  6" 

3,360 

a                 a 

4 

10"  x 

1" 

X 

22'  0" 

Chords  —  flanges. 

2 

u 

X 

12'  0" 

"              " 

1 

a 

X 

6'0" 

1,475 

Bearing-plates  =  4. 

4 

10"  x 

iV' 

x 

6'1" 

254 

End-bracing. 

1 

18"  x 

€ 

X 

9'  6" 

285 

Lugs  on  diagonals  8  —  12"  x  -J" 

1 

9"  x 

X 

18'  8" 

280 

a          a            a                g_   gn    x   y, 

1 

6"   X 

i" 

X 

16'0" 

160 

a         a            a              g_    Q,,    x    y 

1 

9"  x 

i" 

X 

16'  4" 

184 

Web-covers  —  8  —  2'  6" 

1 
1 

'        7"  x 
6"  x 

1" 

X 
X 

2  8" 
9'  7" 

53 
96 

"      gussets  —  4  —  0'  8" 
"      fillers  =  12. 

1 

'         7"  x 

iY' 

x 

1'4" 

10 

End-gussets  =  2. 

1 

8"  x 

r 

X 

2'  9" 

28 

Sway  brace-lugs,  etc. 

1 

'         7"  x 

r 

X 

24'  0" 

210 

Lateral  lugs. 

1 

8"  x 

X 

9'0" 

120 

End-posts,  4  —  2'  3" 

2 

5"  x 

i" 

X 

10'  0" 

83 

End-post  splices,  16  —  1'  3" 

1 

Flat  bar,  3"  x 

r 

X 

12'0" 

60 

Fillers. 

8 

Bars,  angle,  3 

"  X 

3" 

—9-7  Ibs. 

x 

13' 

6' 

Chords. 

8 

u 

II 

a 

X 

28' 

6' 

3,249 

a 

8 

a 

a 

10-8  Ibs. 

X 

6' 

3' 

540 

Diagonals. 

8 

a 

a 

8-2  Ibs. 

X 

6' 

0' 

" 

8 

" 

" 

a 

X 

6' 

3' 

804 

" 

4 

a 

u 

6-8  Ibs. 

X 

6' 

3' 

170 

a 

8 

a 

u 

7'4  Ibs. 

X 

6' 

3' 

370 

a 

8 

a 

a 

6  Ibs. 

x 

6' 

0' 

a 

4 

" 

a 

u 

X 

6' 

3' 

" 

4 

u 

u 

a 

X 

2' 

2' 

Top  chords  outside. 

4 

u 

a 

a 

X 

2' 

6' 

i                            a 

4 

" 

a 

a 

X 

2' 

9' 

i                             a 

2 

u 

a 

« 

X 

8' 

()' 

<                            a 

4 

u 

a 

a 

X 

4' 

9' 

'                inside. 

4 

u 

u 

a 

X 

4' 

6' 

(                          a 

4 

« 

a 

« 

X 

2'  3' 

i                        a 

4 

u' 

U 

a 

X 

2' 

4' 

i                          a 

4 

u 

1 

a 

X 

0' 

r 

End-bracing. 

.       2 

" 

' 

" 

X 

6' 

o' 

Cross-bracing. 

2 

(1 

I 

a 

X 

5' 

5' 

a               a 

8 

u 

1 

a 

X 

4' 

3' 

End  posts. 

16 

" 

' 

" 

X 

7' 

6' 

2,251 

Lateral  bracing. 

8 

Angle-covers,  2£ 

X 

2^_8  Ibs 

X 

no' 

117 

Chords. 

14,159 

Rivets  6  uer 

P.PT1 

h 

"fU1 

15,000 

4 

Cast  bearing-blocks  r«).  1  00  Ihs   

400 

Total  iron  in  poi 

inds 

15,400 

Figs.  955  to  959  are  details  of  portions  of  a  wrought-iron  truss-bridge,  a 
very  good  example  of  usual  American  practice.  Fig.  955  is  a  side  elevation  of 
one  of  the  posts  ;  Fig.  956  a  cross-section  as  far  as  the  first  rail  of  the  road  ; 
Fig.  957  the  lattice  under  side  of  the  top  chord— the  top  is  a  plate.  Fig.  958 
is  a  top  view  of  the  top  chord,  showing  the  lateral  bracing,  consisting  of  a  lat- 
tice box-girder  and  diagonal  rods.  As  the  bridge  is  a  skew,  this  box-girder  is 


ENGINEERING 


429 


430 


ENGINEERING  DRAWING. 


Vl 


\  /i  AX^ 

YV 

Z     -  

J 

1 

'5      .*" 

ENGINEERING  DRAWING. 


431 


n 


not  perpendicular  to  line  of  bridge,  but  parallel  with  abutment.     Fig.  959  is 
the  side  elevation  of  angle  connection  of  end-brace  and  top  chord. 

Figs.  960  to  964  are  illustrations  of  the  landing-bridge  common  at  New 
York  city  ferries.  Fig.  960  is  a  longitudinal  section,  showing  a  section  of  the 
float,  /,  with  its  lever  and  stone  counterpoise  to  balance  the  weight  of  the  bridge, 
the  end  of  which  is  thrown  to  one  side  of  the  float.  Fig.  961  is  the  front  ele- 
vation, and  Fig.  962  the  plan,  one  half  being  planked,  and  one  half  showing- 
framing.  It  will  be  seen  that  there  are  two  chain-barrels,  on  each  side  of  the 
bridge,  worked  by  hand -wheels  ;  on  the  outer  ones  are  the  chains  by  which  the 
boat  is  drawn  up  to  the  bridge  ;  on  the  inner  ones  the  chains  by  which  the 
bridge  is  adjusted  to  the  load  on  the  boat,  and  by  which  a  part  of  the  weight 
of  the  bridge  is  held,  the  upper  ends  of  the  chains  being  attached  to  the  frame 
of  the  overhead.  The  details  (Figs.  963  and  964)  in  section  and  plan  explain 
the  construction  of  the  land-hinge  ;  a  cushion  of  rubber  is  introduced  into 
the  joint  to  modify 
the  shock  caused  by  — n- 

the  boat  striking  the 
bridge,  and  a  flap  of 
wrought-iron  to  cover 
the  joint,  for  protec- 
tion to  travel,  and  se- 
curity from  dirt. 

Piers.— Fig.  965  is 
an  elevation  of  a  pile- 
pier  for  a  bridge.  Ten- 
ons are  cut  on  the  top 
of  the  piles,  and  a  cap 
(a)  mortised  on.  The 
two  outer  piles  are 
driven  in  an  inclined 
position,  and  the  heads 
bolted  to  the  piles  adja- 
cent. The  piles  are  made  into  a  strong  frame  laterally  by  the  planks  b  and  c9 
and  plank-braces  d  d  on  each  side  of  the  piles,  bolted  through.  The  string- 
pieces  of  the  bridge  rest  on  the  cap.  Longitudinal  braces  are  often  used,  their 
lower  ends  resting  on  the  plank  b — which  should  be,  then,  notched  on  to  the 
piles — and  their  upper  ends  coming  together,  or  with  a  straining-piece  between, 
beneath  the  string-pieces,  acting  not  only  as  supports  to  the  load,  but  also  as 
braces  to  prevent  a  movement  forward  of  the  frames.  As  the  tendency  of  a 
moving  train  is  to  push  the  structure  on  which  it  is  supported  forward,  in  rail- 
way-bridges especially,  great  care  is  taken  to  brace  the  structure  in  every 
way — vertically  and  horizontally,  laterally  and  longitudinally.  If  the  plank  c 
be  a  timber-sill,  and  the  piles  beneath  be  replaced  by  a  masonry-pier,  the 
structure  will  represent  a  common  form  of  trestle. 

Fig.  966  is  a  plan  of  one  of  the  stone  piers  of  the  railway-bridge  across  the 
Susquehanna,  at  Havre  de  Grace.  To  lessen  as  much  as  possible  the  obstruc- 
tion to  the  flow  of  the  stream,  it  is  usual  to  make  both  extremities  of  the  piers 


FIG.  965. 


432 


ENGINEERING  DRAWING. 


pointed  or  rounded.    Sometimes  the  points  are  right  angles  ;  sometimes,  angles 
of  60°  ;  often,  a  semicircle,  the  width  of  the  pier  being  the  diameter ;  occa- 
sionally, pointed  arches,  of  which  the  radii  are  the  width  of  the  pier,  the  cen- 
ters being  alternately  in  one  side, 
and   their  arcs   tangent  to  the 
opposite    side.      It   will  be   ob- 
served  (Fig.   966)  that  none  of 
the   stones   break    joint   at  the 
angle — this  is  important  in  op- 
FlG-  966-  posing  resistance   to  drift-wood 

and  ice.     It  is  not  unusual,  in 

very  exposed  places,  to  make  distinct  ice-breakers  above  each  pier,  usually  of 
strong  crib- work,  with  a  plank-slope  like  a  dam,  of  45°,  and  with  a  width 
somewhat  greater  than  that  of  the  pier — a  cheap  structure  as  a  protection  to  an 
expensive  bridge. 

Fig.  967  is  the  plan  and  Fig.  968  the  side  elevation  of  a  pier  of  a  bridge 
across  the  Missouri,  on  the  Northern  Pacific  Kailroad  at  Bismarck,  designed  and 
constructed  by  George  S.  Morison,  C.  E.  In  this  design  both  ends  of  the 
pier  are  rounded,  but  the  upper  extremity  is  extended  beyond  the  main  body 
of  the  pier,  and  the  upper  edge  is  inclined  and  plated  with  iron  between  low 
and  high  water  mark.  This  is  intended 
not  only  to  turn  aside  drift,  but  as  an 
ice-breaker  ;  the  ice,  moving  up  the  in- 
cline, is  broken  by  its  own  weight. 

It  is  now  very  common  in  railroad 
practice  to  construct  wrought-iron  piers, 
as  in  Fig.  969,  of  very  great  height ; 
skeleton-piers,  of  four  or  more  posts, 
adequate  to  sustain  the  load,  with  lat- 
tice girts  and  lateral  rod-bracing. 

Fig.  970  is  a  section  of  the  founda- 
tion of  the  Bismarck  bridge,  showing 
the  construction  of  the  inverted  caisson, 
similar  to  that  used  for  the  Brooklyn 
bridge  pier  and  others.  The  caissons 
are  74'  long,  26'  wide,  and  17'  high  out- 
side ;  the  working-chamber  7  feet  high. 
The  caissons  are  built  of  pine,  sheathed 
with  two  thicknesses  of  3"  oak-plank. 
Above  this  is  crib-work  filled  in  with 
Portland  cement  concrete  ;  a  a  are  the 
air-locks.  The  sand  was  removed  from 
the  caissons  by  water-ejectors. 

Arch  bridges  are  of  stone,  brick,  or  FlG-  969- 

metal ;   the  parts  of   the  arch  exert  a 

direct  thrust  upon  the  abutments,  resisted  by  the  inherent  weight  of  the  latter, 
or  its  absolute  fixed  mass,  as  in  the  case  of  natural  rock  abutments. 


ENGINEERING  DRAWING. 


433 


FIG.  968. 


434 


ENGINEERING  DRAWING. 


FIG.  970. 

Arch  bridges,  in  masonry,  are  arcs  of  circle,  semicircular  (Fig.  972),  segment- 
al  (Fig.  971),  elliptic,  or  described  from  three  or  five  centers  (page  25).  The 
stones  forming  the  arch  are  called  voussoirs,  or  arch-stones  ;  those  forming  the 
exterior  face  are  called  ring-stones,  the  inner  line  of  arch  the  intrados,  exterior 
line  the  extrados.  The  stones  at  the  top,  which  are  those  set  last  and  complete 


the  arch,  are  key-stones.  The  courses  from  which  the  arches  spring  are  called 
skew-backs,  and  the  first  course  the  springing-course.  The  masonry  on  the 
shoulders  of  the  arch  is  called  the  spandrel-courses,  or  spandrel-backing.  The 
weight  at  the  crown  of  a  semicircular  arch  tends  to  raise  the  haunches.  This 
is  counteracted  by  the  spandrel-backing,  and  by  the  earth -load,  which  should 
be  carefully  distributed  on  each  side  of  the  arch. 

To  determine  the  depth  of  the  key-stone,  Rankin  gives  the  following  em- 
pirical rule,  which  applies  very  well  to  most  of  the  above  examples  : 

Depth  at  key,  for  an  arch  of  a  series,  in  feet,  =  V'll  X  radius  at  crown. 
For  a  single  arch,  =  V'12  X  radius  at  crown. 

To  find  the  radius  at  crown  of  a  segmental  arch,  add  together  the  square  of 
half  the  span  and  the  square  of  the  rise,  and  divide  their  sum  by  twice  the  rise — 


ENGINEERING  DRAWING.  435 

Thus,  the  Blackwall  Railway-bridge  has  a  span  of  87  feet,  and  a  rise  of  16 — 
43^5'  + 16*  ^  1892-25  +  256  _ 

2  X  16  ~~32" 

To  find  the  radius  of  an  elliptical  arch,  on  the  hypothesis  that  it  is  an  arch 
of  five  centers  (Fig.  79,  page  25),  the  half-span  is  a  mean  proportional  be- 
tween the  rise  and  the  radius.  Thus,  for  example,  the  Great  Western  Rail- 
way-bridge is  128'  span,  and  24-25'  rise — 

1>42  =  24-25  x  R 

E=ISh169feet- 

To  find  the  depth  of  key-stone,  by  rule  above,  as  in  one  of  a  series — 
d  =  |/17  x  169  =   1/287^=5-33. 

The  depth  of  the  voussoir  is  increased  in  most  bridges  from  the  key-stone  to 
the  springing-course,  but  not  always  ;  it  is  safer  to  increase  the  depth. 

If  an  arch  be  loaded  too  heavily  at  the  crown,  the  lines  of  pressure 
pass  above  the  extrados  of  the  crown,  and  below  the  line  of  intrados  at  the 
haunches,  depressing  the  crown  and  raising  the  haunches,  separating  the 
arch  into  four  pieces,  and  vice  versa  if  the  arches  are  overloaded  at  the 
haunches.  To  prevent  such  effects,  especially  from  moving  loads,  in  con- 
struction the  arches  are  loaded  with  masonry  and  earth,  that  the  constant 
load  may  be  in  such  excess  that  there  will  be  no  dangerous  loss  of  equi- 
librium by  accidental  changes  of  load. 

The  horizontal  thrust  may  be  determined,  according  to  Rankin,  by  the  fol- 
lowing approximate  rule,  which  seldom  errs  more  than  5  per  cent : 

The  horizontal  thrust  is  nearly  equal  to  the  weight  supported  between  the 
crown  and  that  part  of  the  soffit  ivhose  inclination  is  45°> 

This  thrust  is  to  be  resisted  by  the  masonry  of  the  abutment  and  the  earth- 
load  behind  it. 

Thus,  if  Fig.  973  be  a  section  of  an  abutment  of  an  arch,  the  horizontal 
thrust  exerted  at  T  is  resisted  by  the  mass  of  masonry  of  the  abutment ;  the 
tendency  is  to  slide  back  the  abutment  on  its  base 
A  C,  or  turn  it  over  on  the  point  A.  The  sliding 
motion  is  resisted  by  friction,  being  a  percentage, 
say  from  4  to  f  of  the  weight  of  the  abutment  and 
of  half  the  arch  which  is  supported  by  this  base ; 
but,  in  turning  over  the  abutment  on  the  point  A, 
the  action  may  be  considered  that  of  a  lever,  the 
force  T  acting  with  a  lever  T  C  to  raise  the  weight 
of  the  abutment  on  a  lever  A  B  (G  being  the  center 
of  gravity,  and  G  B  the  perpendicular  let  fall  on 
the  base),  and  the  weight  of  half  of  the  arch  on  the  FIG.  973. 

lever  A  C.     That  is,  to  be  in  equilibrium,  the  hori- 
zontal thrust  T  x  T  C  must  be  less  than  the  sum  of  the  weights  of  the  abut- 
ment multiplied  by  A  B,  and  the  weight  of  the  arch  multiplied  by  A  C. 

Skew  bridges  are  those  in  which  the  abutments  are  parallel,  but  not  at  right 
angles  to  the  center  line,  and  the  arches  oblique.  To  construct  these  in  cut 


436 


ENGINEERING  DRAWING. 


stone  requires  intelligence  and  education  both  in  the  designer  and  stone-cutter  ; 
but,  when  the  work  is  laid  full  in  cement,  so  that  the  joints  are  as  strong  as 
the  material  itself,  this  refinement  of  stone-cutting  is  not  necessary.  The  arch 
may  safely  be  constructed  as  a  regular  cylinder  of  a  diameter  equal  to  the  rect- 
angular distance  between  the  abutments,  with  its  extremity  cut  off  parallel  to 
the  upper  line  of  road.  For  such  an  arch  hard-burned  brick  is  the  best  mate- 
rial, the  outer  voussoirs  being  cut  stone. 

In  the  rules  above  given  no  consideration  is  paid  to  the  strength  of  the 
cement  in  which  the  stones  are  bedded.  When  the  cement  is  thoroughly  set,, 
the  structure  is  in  a  measure  monolithic,  and  the  thrust  is  inconsiderable. 


FIG.  974. 


Fig.  974  is  the  elevation  of  one  of  the  stone  arches  of  the  Minneapolis 
Union  Railway  Viaduct,  with  the  timber  centers  on  which  the  arch  was  turned. 
The  arch  is  nearly  semicircular,  97*82  ft.  span,  50  ft.  rise;  width,  28  ft.  ; 
depth  of  arch  at  spring,  40" ;  at  key,  36".  The  piers  are  10  ft.  thick  at  spring- 
ing line  ;  their  up-stream  ends  are  at  angles  to  the  main  body  of  the  piers,  and 
parallel  to  the  thread  of  the  stream.  The  whole  length  is  2,100  ft.,  composed 
of  3  arches  of  40  ft.  span,  16  of  80  ft,  and  4  of  100  ft.  Height  above  water, 
65  ft.  ;  total  height,  82  ft. 

The  centers  were  very  light  frames.  5  to  each  arch ;  the  chords,  timber 
arches,  and  ties  were  each  12"  X  12",  the  central  braces  10"  X  10",  and  the 
shorter  side-braces  8"  X  8"  ;  the  bolts,  single,  H"  diameter. 

The  bridge  was  constructed  after  the  designs  and  under  the  direction  of 
Charles  C.  Smith,  C.  E.,  Chief  Engineer  of  the  St.  Paul,  Minneapolis  and 
Manitoba  Railway,  and  is  an  example  of  a  very  economical  and  stable  con- 
struction. The  piers  are  of  Minnesota  granite,  but  above  springing  line  the 
masonry  is  of  magnesian  limestone.  It  was  commenced  in  February,  1882, 
and  completed  in  November,  1883. 


ENGINEERING  DRAWING. 


437 


LOCATION. 

Material. 

Form  of  arch. 

Span. 

Rise. 

Depth 
at 
crown. 

Depth 

at 
spring 

Manchester  and  Birmingham  Railroad  

u              u                    a                      « 

London  and  Brighton  Railroad                . 

Brick. 

u 
U 

Semicircular. 

U 
it 

18 
63 
30 

9 
31-6 
15 

1-6 
3 
1'6 

Unif'rm 

u 
2-3 

u        "     Blackwall       " 

„ 

SeTnental 

87 

16 

4'U 

Unif'rm 

Great  Western  Railroad  

u 

Elliptical. 

128 

24-3 

5 

7'H 

Chestnut  Street  (Philadelphia)  Railroad.  .  . 
High  Bridge,  Harlem  River,  New  York  .  .  . 
St.  Paul,  Minneapolis  &  Manitoba  Railroad 
(largest  arch),  at  Minneapolis  

u 

Stone. 

u 

Segmental. 
Semicircular. 

Segmental. 

60 
80 

97'8 

18 
40 

50 

2-6 

2-8 

3 

3-4 

Cabin  John  Washington  Aqueduct  

u 

Elliptical 

220 

57-3 

4'2 

Lickinf  Aqueduct  and  Ohio  Canal        .    .  . 

u 

u 

90 

15 

2*10 

Monocacy       "         

u 

u 

54 

9 

2*6 

Hutcheson     u                         

a 

Segmental 

79 

13'6 

3'6 

4'6 

Chcmin  du  Fer  du  Nord  sur  1'Oise 

U 

u 

82'5 

13*5 

4'6 

D'En^hien  Railroad  du  Nord  

u 

Semicircular. 

24-4 

12'2 

1'4 

Du  Crochet  Railroad  

M 

it 

13-2 

6*6 

1-7A 

Experimental  arch,  designed  and  built  by 
M   Vaudray  Paris   .      ... 

Se  "mental 

124 

6*11 

A    '2 

2*8 

3'7 

The  arch  last  in  the  list  was  a  very  bold  specimen  of  engineering,  built  as 
an  experiment,  preliminary  to  the  construction  of  a  bridge  over  the  Seine.  It 
was  made  of  cut  stone,  laid  in  Portland  cement,  with  joints  of  f",  and  left  to 
set  four  months ;  the  arch  was  12'  wide  ;  the  centers  rested  on  posts  in  wrought- 
iron  boxes  filled  with  sand,  and,  as  the  centering  was  eased  by  the  running  out 
of  the  sand,  the  crown  came  down  T6¥" ;  the  joints  of  one  of  the  skew-backs 
opening  10'T00/!r  during  the  first  day,  it  came  down  y^h/'.  It  was  then  loaded 
with  a,  distributed  weight  of  300  tons  ;  under  this  load  the  crown  settled  -f^" 
more.  Since  then  nothing  has  stirred,  although  it  was  afterward  tested  by 
allowing  five  tons  to  fall  vertically  1'  6"  on  the  roadway  over  the  key-stone. 
This  bridge  will  not  come  within  any  of  the  rules  laid  down  for  other  construc- 
tions. It  will  be  observed  that  the  rise  is  about  -fa  the  span,  although  the 
usual  practice  for  segmental  and  elliptical  arches  is  more  than  £,  or  within  the 
limits  of  and 


FIG.  976. 

In  suspension-bridges  the  platform  of  the  bridge  is  suspended  from  cables, 
or  chains,  the  ends  of  which  are  securely  anchored  within  the  natural  or  arti- 
ficial abutments. 


438 


ENGINEERING  DRAWING. 


The  curve  of  a  suspended  chain  is  that  known  as  the  catenary,  and,  if  the 
whole  weight  of  the  structure  were  in  the  chain  itself,  this  would  be  the  curve 
of  the  chains  of  a  suspension-bridge  ;  but,  as  a  large  part  of  the  weight  and 
the  whole  of  the  loading  lies  in  the  platform,  the  curve  assimilates  to  that  of  a 
parabola,  and  in  all  calculations  it  is  so  regarded. 

Let  Fig.  976  represent  a  suspension-bridge,  in  which  A,  B,  C,  are  points  in 
a  parabolic  curve. 

Rule. — Add  together  four  times  the  square  of  the  deflection  (E  B)2  and  the 
square  of  half-span  (AE)2,  and  take  the  square  root  of  this  sum  ;  multiply  this 
result  by  the  total  weight  of  one  chain  and  all  that  is  suspended  from  it,  in- 
cluding the  distributed  load,  and  divide  this  product  by  four  times  the  deflec- 
tion (E  B)  of  the  cable  at  the  center,  and  the  result  will  be  the  tension  on  one 
chain,  at  each  point  of  support,  A  and  0.  The  angle  made  by  the  chain  at 
the  point  of  support,  viz.,  angle  POL  and  the  angle  of  the  back-stays,  or  con- 
tinuation of  the  chain  (angle  L  C  N)  should  be  equal  to  each  other,  in  order 
that  there  be  no  tendency  to  overset  the  tower  C  L  and  A  F. 


BRIDGES. 

Main 
spans. 

Deflection 
of  chain  or 
cable. 

No.  of 
chains  and 
cables. 

Total  effective 
section  of  cable  in 
square  inches. 

Mean  weight 
of  cable  per  foot  of 
span  (pounds). 

Fixed  load 
per  foot  of 
span  (Ibs.). 

Breadth  of 
platform 
in  feet. 

Menai    

570 

43 

16 

260 

880 

28 

Chelsea.      .  .  . 

348 

29 

4 

230 

767 

47 

Pesth        

666 

47'6 

4 

507 

1  690 

9  892 

46 

Bamberg  

211 

14-1 

4 

40-2 

137 

1,581 

30-5 

Freyburg  

870 

63 

4 

49 

167 

760 

21-25 

Niagara  Falls. 

821 

54  and  64 

4 

241-6 

820 

2,032 

24 

Cincinnati  .  .  . 

1,057 

89 

2 

172-6 

516 

2,580 

36 

Brooklyn  .... 

1,595 

BOILER-SETTING. 

Fig.  977  is  a  longitudinal  section,  Fig.  978  a  plan  with  section  of  wall,  and 
Fig.  979  an  elevation  half-front  and  half-sectional  of  a  boiler  and  setting  as 
recommended  by  the  Hartford  branch  of  the  Hartford  Steam-Boiler  and  In- 
spection Company,  showing  the  interior  bracing,  steam  and  water  connections, 
and  brick-work.  There  are  ten  braces  in  each  head,  secured  to  pieces  of  T-iron, 
placed  radially,  as  shown  in  dotted  lines  (Fig.  979).  The  feed-pipe  is  through 
the  front-head,  just  above  the  line  of  tubes,  extending  to  the  back  of  the  boiler, 
with  a  perforated  branch  across  it,  that  the  water  may  be  warmed  in  its  passage 
and  distributed.  The  front  is  a  projecting  cut-away  front,  the  boiler-head 
being  nearly  on  a  line  with  the  front  below,  diiferent  from  that  given  in  Fig. 
768,  where  the  lower  part  of  the  shell  projects  beyond  the  head  of  the  boiler, 
and  the  cast-iron  front  covers  the  end.  The  doors  giving  access  to  the  tubes 
are  usually  semicircular,  and  hung  on  the  top  diameter,  but  it  will  be  found 
more  convenient  to  form  them  in  two  quadrants,  and  hung  so  as  to  move  hori- 
zontally. The  boilers  are  to  be  protected  against  radiation  by  a  covering  of 
ashes,  or  a  brick  arch,  resting  on  the  side-walls.  My  own  practice  is  to  ninke 
the  boiler  without  lugs  to  support  it  on  the  side-walls,  but  to  hang  the  boiler 


ENGINEERING  DRAWING. 


439 


13  2' 


from  wrought-iron  cross-bars,  resting 
on  the  top  of  the  side-walls,  and  put- 
ting small  bars  across  just  above  the 
top  of  the  boiler,  to  roof  over  with 
sheet-iron  and  fill  above  with  ashes, 
leaving  the  spandrels  as  hot-air  spaces. 
It  will  be  observed  (Fig.  978)  that 
the  manhole-frame  is  riveted  to  the  in- 
side of  the  boiler ;  frequently  it  is  on 
the  outside.  For  most  positions  I  pre- 
fer that  the  manhole  should  be  placed 
in  the  back-head,  as  easier  of  access, 
and  in  my  form  of  cover  there  is  no 
disturbance  of  ashes  for  access  to  the 
manhole.  It  is  often  well  to  make  the 
blow-off  pipe  a  circulating  pipe  by 


FIG.  979 


440 


ENGINEERING  DRAWING. 


00 
000 
000 
000 

ooo 
ooc 
ooo 
ooo 
ooo 
ooo 


oooo 
oooo 
oooc 
oooo 
oooo 
oooo 
oooo 
oooo 
oooo 
oooo 


ooo  oooo 
ooo  oooo 
ooo  oooo 


ooo 

ooo  oooo 

000  OOOO 
000  OOOO 
000  OOOO 

ooo  oooo 

00  OOOO 


ENGINEERING  DRAWING. 


44:1 


Fm.  982. 


FIG.  985. 


442 


ENGINEERING   DRAWING. 


FIG.  987. 


FIG.  988. 


connecting  an  inch  pipe 
inside  the  valve  with  the 
upper  water-space  of  the 
boiler. 

Fig.  981  is  a  longitu- 
dinal section,  and  Fig. 
980  half  front  elevation 
and  half  cross-section  of 
a  class  of  boilers  usual- 
ly designated  as  marine 
boilers,  but  largely  used 
at  the  Philadelphia  Wa- 
ter-Works. The  fire- 
boxes and  ash-pits  are 
contained  within  the 
body  of  the  boiler  ;  it  is 
set  on  a  cast-iron  or  brick 
base,  and  the  shell  is 
covered  with  some  prep- 
aration of  plaster  or  hair- 
felt  clothing.  The  front 
smoke-box  is  of  wrought- 
iron,  and  similar  to  that 
shown  in  Fig.  977. 

Locomotive-boilers 
are  used  as  stationaries, 
and  are  set  like  the  pre- 
ceding, but  with  some 
non-conducting  cover- 
ing. The  protection  of 
all  parts  of  boilers  and 
steam-pipes  exposed  to 
the  air  by  some  cover  of 
a  non-conducting  mate- 
rial adds  much  to  econ- 
omy in  the  consumption 
of  coal  and  dryness  of 
steam. 

Fig.  982  is  a  vertical 
section  of  a  chimney  at 
the  Eidgewood  Pumping- 
engine  House,  Brooklyn, 
K  Y.,  and  Fig.  983  an 
elevation  at  the  point 
where  the  square  base  is 
changed  into  an  octago- 
nal. 


ENGINEERING  DRAWING. 


443 


FIG.  991. 


4:4:4:  ENGINEERING  DRAWING. 

Fig.  984  is  a  section  of  the  shaft  at  a  b,  but  the  flue  should  have  been  repre- 
sented circular. 

Fig.  985  is  a  vertical  section  of  a  chimney  attached  to  an  English  gas-house, 
taken  from  "  Engineering,"  with  a  uniform  flue  and  shell,  additional  strength 
being  given  by  the  buttresses  shown  in  section  at  c  d  (Fig.  986).  No  inde- 
pendent flue  inside  is  shown,  but  it  is  desirable,  as  it  can  freely  expand  with 
the  heat,  without  affecting  the  outer  shell. 

Fig.  988  is  the  cross-section  of  a  buttressed  chimney  at  100  feet  above  base, 
built  for  the  Calumet  &  Hecla  Mining  Company,  and  designed  by  E.  D. 
Leavitt,  Jr.,  M.  E.  The  whole  height  of  the  chimney  is  150  feet.  The  but- 
tress walls  are  16"  and  12"  thick,  that  of  the  body  12"  and  8",  and  of  the  cen- 
tral flue  8"  and  4",  offsetting  into  each  other  by  1"  oifsets  ;  the  taper  is  4  inches 
to  10  feet  on  each  side.  Fig.  987  is  a  half  elevation  and  half  section  of  the 
cap  and  the  cover  of  the  interior  flue  by  which  its  expansion  is  permitted. 

Fig.  989  is  a  sectional  elevation  of  a  chimney  160  feet  high,  from  John  T. 
Henthorne,  M.  E.,  with  a  cross-section  (Fig.  990)  midway  of  the  height. 

Figs.  991  and  992  are  sectional  elevation  and  cross-section  of  a  chimney  of 
my  own  design  and  construction.  The  buttresses  supporting  the  central  flue 
are  inside  the  chimney.  The  diameter  of  the  flue  is  4  feet,  and  the  height 
about  100  feet. 

It  has  not  been  my  practice  to  build  high  chimneys — 100  feet  is  usually  suf- 
ficient— but  they  should  extend  above  surrounding  houses,  woods,  and  hills, 
which  are  near  enough  to  influence  the  draught.  For  chimneys  of  this  height 
an  area  of  chimney-flue  of  one  square  inch  for  every  pound  of  anthracite  coal 
burned  per  hour  on  the  grate  has  been  found  to  answer  well  in  practice.  For 
chimneys  less  than  this  height,  it  is  well  to  increase  the  section,  and  perhaps 
reduce  for  higher  chimneys. 

Chimneys  are  constructed  of  various  sections,  sometimes  uniform  through- 
out their  length,  sometimes  tapering  at  the  top,  and  sometimes  bell-mouthed  ; 
all  answer  the  purpose.  The  great  point  to  be  observed  is,  that  there  be  no 
abrupt  changes  of  section  or  direction,  either  in  the  main  flue  or  its  connec- 
tions, and  that  they  be  carried  well  above  all  disturbing  causes. 

ON   THE   LOCATION    OF   MACHINES. 

The  construction  of  buildings  for  mills  and  manufactories  (if  any  aesthetic 
effect  is  intended)  is  usually  left  to  the  architect,  but  the  necessities  of  the 
construction,  the  weights  to  be  supported,  and  the  space  to  be  occupied,  must 
be  supplied  by  the  mechanical  engineer  or  millwright. 

In  the  arrangement  of  a  manufactory  or  workshop,  it  is  of  the  utmost  im- 
portance to  know  how  to  place  the  machinery,  both  as  to  economy  of  space  and 
also  of  working.  Where  a  new  building  is  to  be  constructed  for  a  specific  pur- 
pose of  manufacture,  it  will  be  found  best  to  arrange  the  necessary  machines 
as  they  should  be,  and  then  build  the  edifice  to  suit  them.  For  defining  the 
position  of  a  machine,  the  space  it  occupies  in  plan  and  elevation,  the  position 
of  the  driven  pulley  or  gear,  of  the  operative,  and  spaces  for  the  working  and 
access  to  parts,  are  required.  To  illustrate  this  subject,  take  a  two-story  weav- 
ing-room, of  which  Fig.  993  is  an  elevation,  and  Fig.  994  a  plan. 


ENGINEERING  DRAWING. 


445 


Lay  down  the  outlines  of  an  interior  angle  of  the  building,  and  dot  in,  or 
draw  in  red  or  blue,  the  position  and  width  of  beams.  This  last  is  of  impor- 
tance, as  it  will  be  observed  (Fig.  993)  that  no  driving-pulley  can  come  beneath 


V/////X///M  /IVM/M/M     !44^yy>y///t^J^^^^ 


irt\  J  1  1  j  j  ]  i  I 


FIG.  993. 


the  beam,  and  also  that  this  is  the  position  for  the  hanger.  Lay  off  now  the 
width  of  the  alleys  and  of  the  machines.  The  first  alley,  or  nearest  the  side- 
wall,  is  a  back  alley  ;  that  is,  where  the  operative  does,  not  stand,  and  so  on 
alternate  alleys.  Draw  the  lines  of  shafting  central  to  the  alleys,  as  in  this 


446 


ENGINEERING   DRAWING. 


FIG.  994. 


ENGINEERING  DRAWING.  447 

• 

position  the  belts  are  least  in  the  way.  One  operative  usually  tends  four  looms  ; 
they  are  therefore  generally  arranged  in  sets  of  four,  two  on  each  side  of  the 
main  alley,  where  the  operative  stands  ;  the  twos  are  placed  as  close  to 
each  other  as  possible,  say  one  inch  between  the  lays,  a  small  cross-alley  being 
left  between  them  and  the  next  set.  Lay  off  now  the  alley  necessary  at  the  end 
of  the  room,  and  space  off  the  length  of  two  rows  of  looms  with  alleys  at  the 
end  of  alternate  looms,  and  mark  the  position  of  the  pulleys.  It  will  be  ob- 
served that  looms  are  generally  rights  and  lefts,  so  that  the  pulleys  of  both 
looms  come  in  the  space  where  there  is  no  alley.  Should  the  pulley  come  be- 
neath a  beam,  the  loom  must  be  either  moved  to  avoid  it,  or  the  pulley  may  be 
shifted  to  the  opposite  end  of  the  loom.  Parallel  with  the  pulleys  on  the  looms 
draw  the  driving-pulleys  on  the  shafts,  that  is,  k  parallel  with  &,  b  with  b,  f 
with/,  and  so  on.  Proceed  now  to  draw  the  third  and  fourth  row  of  looms, 
since  the  second  and  third  rows  are  driven  from  the  same  shaft ;  if  they  are 
placed  on  the  same  line,  it  will  be  impossible  to  drive  both  from  the  same  end, 
and,  as  this  is  important,  we  move  the  third  row  the  width  of  the  pulley  b,  and, 
for  the  sake  of  uniformity,  the  fourth  row  also.  Lay  off  now  the  length  of 
looms  and  position  of  pulleys  as  before,  and  parallel  with  the  pulleys  the  driving- 
pulleys  on  the  shaft,  that  is,  c  against  c,  ^against  g,  and  so  on.  Having  in 
this  way  plotted  in  all  the  looms,  every  alternate  set  being  on  a  line  with  the 
third  and  fourth  row,  proceed  now  to  lay  down  the  position  of  the  looms  in 
the  floor  above  ;  and  since  for  economy  of  shafting  it  is  usual  to  drive  from  the 
lines  in  the  lower  rooms,  to  avoid  errors,  interference  of  belts  and  pulleys,  it  is 
usual  to  plot  the  upper  room  on  the  same  paper  or  board  as  the  lower  room, 
using  either  two  different  colored  inks,  or  drawing  the  machines  in  one  room 
in  deep  and  in  the  other  in  light  line,  as  shown  in  Fig.  994.  If  the  width  of 
the  rooms  is  the  same,  the  lateral  lines  of  looms  and  alleys  are  the  same,  and 
it  is  only  necessary,  therefore,  to  fix  the  end  lines.  Now,  as  the  first  loom  in 
the  outer  row  of  looms,  in  the  lower  room,  occupies  for  its  belt  the  position  k 
on  the  shaft,  the  loom  in  the  upper  room  must  be  moved  either  one  way  or  the 
other  to  avoid  this  ;  thus  the  position  i  of  the  pulley  on  the  loom  must  be  made 
parallel  to  the  pulley  i  on  the  shaft,  so  in  the  other  looms  a  to  a,  e  to  e,  d  to 
d,  and  h  to  h. 

Besides  the  plan,  it  is  often  necessary,  and  always  convenient,  to  draw  a 
sectional  elevation  (as  in  Fig.  993)  of  the  rooms,  with  the  relative  positions  of 
the  driving-pulleys  and  those  on  the  machines,  to  determine  suitably  the  length 
of  the  belts,  and  also  to  see  that  their  position  is  in  every  way  the  most  con- 
venient possible.  In  the  figure,  one  of  the  lower  belts  should  have  been  a 
cross-belt,  and  one  of  the  upper  ones  straight  :  now,  had  the  belts  to  the  second 
row  of  looms  in  the  upper  story  been  drawn  as  they  should  have  been,  straight, 
the  belt  would  have  interfered  a  little  with  the  alley,  and  it  would  have  been 
better  to  have  moved  the  driving-shaft  a  trifle  toward  the  wall. 

From  this  illustration  of  the  location  of  machines,  knowing  all  the  require- 
ments, in  a  similar  way  any  machinery  may  be  arranged  with  economy  of  space, 
materials,  power,  and  attendance.  These  last  two  items  are  of  the  more  im- 
portance as  they  involve  a  daily  expense,  where  the  others  are  almost  entirely 
in  the  first  outlay. 


448 


ENGINEERING  DRAWING. 


ENGINEERING   DRAWING. 


449 


Machine  Foundations. — Figs.  995,  996,  and  997  are  side  and  end  elevation, 
and  plan,  of  the  foundation  of  the  stationary  steam-engine.  F  is  the  cast-iron 
frame  or  bed-plate  of  the  engine  ;  B  the  granite  bed  of  engine,  or  coping  of 
foundation;  P  the  stone  or  brick  pier,  laid  full  in  cement.  The  sides  and  sur- 
faces of  granite  exposed  are  usually  fine-hammered,  the  upper  bed  or  build  to 
receive  the  engine-frame,  hammer-dressed  and  set  level.  Strong  wrought-iron 
bolts  pass  through  frame,  bed,  and  pier,  with  nuts  at  each  end,  and  the  whole 
is  strongly  bolted  together.  Pockets  are  left  in  the  pier  near  the  bottom  for 
access  to  nuts,  and  these  pockets  are  covered  by  granite  caps  or  iron  plates. 

Few  stationary  steam-engines  are  now  built  with  bed-plates  extending  the 
whole  length  of  the  engine,  but  the  illustration  is  applicable  to  the  partial 
plates  supporting  the  cylinder  and  pillow-block,  and  to  engines  and  machines 
for  which  heavy  foundations  are  necessary.  It  is  not  an  uncommon  practice 
now,  instead  of  granite  caps,  to  use  timber,  as  cushioning  the  shocks  and  blows 
incident  to  most  machinery. 

Tunnels. — Figs.  998  to  1007  are  illustrations,  with  description,  taken  from 
"  Tunneling,"  a  standard  work  on  this  subject  by  H.  S.  Drinker. 

Figs.  998  to  1003  illustrate  the  principles  of  timbering  applied  to  driving  a 
gallery  through  running  material.  Figs.  998  and  999  are  parts  of  the  construc- 


ipqiuip  ooaros  aajoimnj 
FIG.  998. 


FIG.  999. 


tion  on  a  large  scale,  with  the  technical  names  of  the  parts.  Each  frame  is 
called  a  timber-set.  Suppose  a  leading  set  (Figs.  1000  and  1001)  is  in  place, 
close  to  the  face,  and  that  the  leading  ends  of  the  poling-boards  resting  above 
this  leading  set  are  held  up  from  the  collar  by  wedges  sufficiently  high  to  allow 
the  insertion  of  the  new  poling-boards.  In  Fig.  1001  the  sets  e  e,  standing  mid- 
way between  the  front  and  the  hind  ends  of  the  poling-boards,  serve  as  middle 
sets  between  the  main  sets  d  d.  By  turning  to  the  plan  (Fig.  1003)  of  a  gal- 
lery thus  timbered  it  will  be  seen  that,  owing  to  the  fact  that  the  site-poling 
has  also  to  be  wedged  out  at  its  leading  end,  just  as  the  roof-poling  is  wedged 
up,  therefore  the  space  to  be  filled  across  the  top  by  the  roof -poling  is  wider 
over  a  front  main-set  than  over  a  back  one.  Owing  to  this  fact,  the  two  outer 


2'J 


4:50 


ENGINEERING  DRAWING. 


top  poling-boards,  as  shown  in  Fig.  998,  are  made  wider  at  their  leading  ends 
than  at  their  back  ends.  Now,  to  begin  inserting  the  roof-poling,  the  miners, 
at  either  corner  of  the  face,  remove  the  extreme  end- wedges  between  the  collars 
and  the  poling,  and  into  this  space  the  new  poling-boards  (i.  e.,  the  ones  shown 
in  Fig.  998)  that  are  wider  at  their  leading  ends  are  driven.  But,  though  the 


FIG.  1000. 


FIG.  1001. 


wedges  between  the  collar  and  the  poling-boards  serve  well  enough  to  keep  back 
the  material,  it  would  be  dangerous  thus  to  take  any  of  them  out  were  there  no 
other  guard  for  the  poling,  as  the  board  just  above  the  wedge  removed  would 
be  pressed  down  ;  a  run  might  also  be  started,  and  all  the  other  wedges  forced 

out,  when  the  poling-boards  would  snap 
down  on  the  leading  collar,  and  per- 


FIG.  1002. 


FIG.  1003. 


haps  break  off  ;  in  any  event,  it  would  be  a  matter  of  great  trouble  to  get  them 
wedged  up  again.  In  order  to  guard  against  this  trouble,  a  cross-board  or 
plank  a  (Fig.  999)  is  placed  just  under  the  poling-boards,  and  over  the  wedges. 
Then,  when  one  wedge  is  removed,  this  cross-connection  holds  in  place  the 
poling-board  that  is  immediately  above  the  wedge  removed,  until  the  new  board 


ENGINEERING 


FIG.  1004. 
(Section  of  Fig.  1006,  through  A  B,  looking  west.) 

HOOSAC  TUNNEL. 
Timbering  and  arching  through  soft  ground  at 'the  West  End.    Scale,  11'  =  1* 


FIG.  1005. 


452 


ENGINEERING  DRAWING. 


West. 


FIG.  1006. 

HOOSAC  TUNNEL. 
West  End.     Scale,  11'  =  1. 


FIG.  1007. 


ENGINEERING  DRAWING.  453 

can  be  put  in  ;  it  also  stays  the  tendency  to  any  general  movement.  The  new 
poling-board  being  inserted,  it  is  now  driven  ahead  six  or  twelve  inches,  and 
then  temporarily  stayed  by  wedges,  b  (Fig.  1001).  The  corner  roof-polings 
being  thus  in  place,  the  middle  ones  (Fig.  998)  are  similarly  inserted.  Then 
the  top  retaining-board  in  the  face  is  cut  out,  and  the  material  allowed  to  flow 
into  the  heading  through  the  space.  As  room  is  thus  given  ahead,  the  poling- 
boards  are  gradually  driven  forward,  say  24  or  30  inches,  or  about  half  the 
length  of  a  board,  supposing  they  are  5  feet  long.  Whenever  they  are  thus 
tapped,  the  wedges  I  (Fig.  999)  must  be  loosened,  and  then  tightened  again 
after  the  driving.  The  side-poling  is  similarly  thus  advanced ;  and  we  must 
bear  in  mind  that,  as  space  is  gained  ahead,  it  must  be  protected  by  new  face- 
boarding,  stayed  by  stretchers.  Thus  the  work  can  be  gradually  carried  down 
to  the  floor  of  the  heading,  by  successively  taking  out  the  face-boards.  Often 
the  floor  of  the  gallery  also  has  to  be  planked,  and,  in  very  extreme  cases,  to  be 
poled  similarly  to  the  roof  and  sides. 

We  now  have  reached  the  point,  shown  in  Fig.  1002,  where  the  new  poling- 
board  has  been  inserted  for  its  half-length.  During  this  operation  the  boards 
have  been  held  in  place  by  the  double  support  oifered  by  a  and  b  (Fig.  1002). 
The  face  retaining- boards  are  kept  back  by  a  vertical  plank  laid  across  them, 
and  stayed  by  stretchers.  On  this  newly-excavated  chamber  the  outside  pressure 
will  be  great,  especially  acting,  as  it  does,  on  the  front  half  length  of  the  poling- 
board  c  a,  and,  if  the  remaining  work  is  not  rapidly  executed,  the  front  ends  of 
the  boards  may  be  snapped  beyond  a  ;  then,  if  it  were  attempted  to  drive  the 
remaining  portion  of  the  board  on,  as  soon  as  its  back  end  left  b  it  would  snap 
between  a  and  b.  A  middle  set  is  therefore  required  at  once.  The  middle  set 
being  in  position,  the  work  of  excavating  the  face  can  be  proceeded  with  as 
before.  The  face-boards  are  removed,  one  by  one,  from  top  to  bottom,  and 
the  polings  are  driven  in  to  their  full  length  ;  then  in  the  new  length  ahead  the 
next  main  set  is  erected. 

Such  are  the  general  principles  of  head  ing-driving  thro  ugh  running  ground, 
or  sheet-piling  in  tunneling. 

Figs.  1004  to  1007  show  the  English  system  of  bar-timbering,  as  used  at 
the  Hoosac  Tunnel  for  the  soft  ground  at  the  west  end.  The  material  was  of 
the  worst  character,  and  was  exceedingly  difficult  to  drive  through.  Figs. 
1004  and  1005  are  cross-sections,  the  one  looking  west  from  A  B,  the  other 
east.  Fig.  1006  is  a  longitudinal  section.  Fig.  1007  is  a  cross-section  of  the 
tunnel  as  completed  with  an  invert,  and  the  bars  not  drawn  but  bricked  in. 

Railway  Stock.— Figs.  1008  and  1009  are  the  elevation  and  plan  of  a  stand- 
ard box-car  of  the  New  York  Central  and  Hudson  River  Railroad. 

Figs.  1010  to  1013  are  the  plan  and  elevations  of  the  truck  for  the  same  car. 

Figs.  1014,  1015,  and  1016  are  end-elevations  and  cross-sections,  Figs. 
1017  and  1019  longitudinal  sections,  and  Fig.  1018  plan  of  a  standard  passen- 
ger-car of  the  Pennsylvania  Railroad. 

Figs.  1020  to  1023  are  elevations,  in  full  and  parts,  and  Fig.  1024  a  plan  of 
the  trucks  of  the  above  car. 

In  the  figures,  both  of  standard  box  and  passenger  cars,  the  elevations  and 
plans  are  usually  broken,  to  show  the  construction.  When  the  two  sides  or 


454: 


ENGINEERING   DRAWING. 


ENGINEERING  DRAWING. 


455 


—  B 


or  side  Door- 


!•< -Spring  Plank  2'-IO'/z" 


{< Transom  3  -3  " -x Swing  Bolster.  2  -10 fc" I *, 


Oentercf  Sffing  2-4"    •  - ->J 


Between  Centers  of  Journal  Bearing  6-3 


FIG.  1013.          'rtr Axle  e-'55/« 

Ervd  Elevation. 


456 


ENGINEERING  DRAWING. 


two  ends  of  a  car  or  truck  are  similar,  it  has  not  been  considered  necessary  to 
show  both,  but  complete  the  figure,  with  a  section  of  the  other  part,  through  a 
different  plane. 

STANDARD   PASSENGER  CAR  OF  THE   PENNSYLVANIA  RAILROAD 


ENGINEERING  DRAWING. 


457 


TRUCK  OF  PENNSYLVANIA  RAILROAD   STANDARD  PASSENGER  CAR. 


^Er£g3^'  ^"H-  ^    -^-i      1    .  . 


Fio.  1024. 


458 


ENGINEERING  DRAWING. 


and  technical  names  of  similar  parts 


The  following  letters  of  reference 
apply  equally  to  all  the  figures  : 

a,  Sill. 

a',  End-sill. 

6,    Intermediate  floor-timbers. 

6',  Center  floor-timbers. 

c,  Sill  knee-iron  or  strap. 

d,  Body  bolster. 

e,  Body  bolster  truss-rod. 
/,   Truck  side-bearing. 

<7,    Center  plate,  body  or  truck. 

A,    Check-chain  on  the  truck,  hooking  into 

A',  Check-chain  eye  on  the  car. 

i,     Body  truss-rod. 

i',   Body  truss-rod  queen-post. 

y,    Cross-frame  tie-timber. 

The  Wave-line  Principle  of  Ship- Construction,  from  Russell's  "Naval  Archi- 
tecture."— The  general  doctrines  arrived  at  by  J.  Scott  Russell,  F.  R.  S.,  from 
numerous  and  long-continued  experiments  and  practical  tests,  is  "  that  the 
form  of  least  resistance  for  the  water-line  of  the  bow  is  horizontally  the  curve 
of  versed  sines,  and  that  the  form  of  least  resistance  for  the  stern  of  the  vessel 
is  the  cycloid  ;  and  you  can  either  adopt  the  said  cycloid  vertically  or  horizon- 
tally, or  you  can  adopt  it  partly  vertically  and  partly  horizontally,  according 
to  the  use  of  the  vessel  or  the  depth  of  water. " 

"That  the  length  of  entrance,  or  fore  body,  should  be  f,  and  that  of  the 
run,  or  after  body,  f . " 

"When  it  is  required  to  construct  the  water-lines  of  the  bow  of  a  ship  of 
which  the  breadth  and  the  length  of  the  bow  are  given,  so  as  to  give  the  vessel 


Draw-bar. 

Journal-box. 

Pedestal. 

Pedestal  tie-bar. 

Pedestal  stay-rod. 

Pedestal  arch-bar. 

Pedestal  inverted  arch-bar. 

Transom. 

Truck  bolster. 

Spring-plank. 

Swing-hanger. 

Safety-beam. 

Equalizing-bar. 


FIG.  1025. 


the  form  of  least  resistance  to  passage  through  the  water,  or  to  obtain  the  high- 
est velocity  with  a  given  power  :  Take  the  greatest  breadth,  M  M  (Fig.  1025), 
on  the  main  section  of  construction  at  midship-breadth,  and  halve  this  breadth, 
M  0  ;  at  right  angles  to  M  M  at  0  draw  the  center  line  of  the  length  of  the 
bow,  0  X  ;  on  each  half -breadth  describe  a  half -circle,  dividing  its  circumfer- 


ENGINEERING  DRAWING. 

ence  into,  say,  eight  equal  parts.  Divide  the  length  0  X 
into  the  same  number  of  equal  parts.  The  divisions  of  the 
circle,  reckoned  successively  from  the  extreme  breadth,  indi- 
cate the  breadths  of  the  water-line  at  the  successive  corre- 
sponding points  of  the  line  of  length.  Through  the  divis- 
ions of  the  circles  draw  lines  parallel  to  0  X,  and  through 
the  divisions  of  0  X  lines  parallel  to  M  M.  These,  inter- 
secting one  another,  show  the  successive  points  in  the  re- 
quired water-line.  The  line  traced  through  all  these  points 
is  the  wave  water-line  of  least  resistance  for  a  given  length 
of  bow  and  breadth  of  body. " 

To  construct  the  water-lines  of  the  after  body  or  run  of 
a  ship  (Fig.  1027),  the  mid-section  (Fig.  1026)  being  given  : 
The  bow  is  constructed  as  in  Fig.  1025,  but  the  divisions  are 
12  on  the  center  line  ;  for  the  run  lay  off  8  divisions,  each 


459 


FIG.  1026. 

equal  to  those  of  the  bow  ;  divide  the  half  circle  into  8  equal 
parts,  and  draw  chords  to  these  divisions  from  0  to  1,  2,  3,  4. 
From  the  point  1  on  the  center  line  lay  off  an  inclined  line 
equal  and  parallel  to  the  chord  0 1  ;  the  point  1'  will  be  in 
the  water-line.  In  the  same  way  from  the  point  2  draw  an 
inclined  line  parallel  and  equal  to  the  chord  0  2,  for  2',  and  J 
determine  in  the  same  way  the  points  3',  4',  5',  6',  7'.  The 
other  circles  drawn  in  the  figure  are  described  on  semi- 
diameters  of  the  mid-section  at  different  levels,  and  the 
points  of  their  wave-lines  are  determined  on  the  same  in- 
clined lines  1 1',  2  2',  but  the  lengths  are  those  of  the 
chords  of  the  different  circles.  In  Fig.  1026,  the  elevations 
of  the  mid  body,  the  curved  lines  of  sections  are  projected 
from  the  plan. 

Fig.  1028  is  a  body  plan  of  a  vessel  adapted  to  speed ; 
Fig.  1029  of  one  adapted  to  freight. 

"  To  determine  the  after  body  it  is  expedient  to  construct 
a  vertical  wave-line  on  the  run  as  well  as  a  horizontal  one, 
and  in  designing  shallow  vessels  to  give  more  weight  to  the 
vertical  wave-line." 

"  The  wave  system  destroys  all  idea  of  any  proportion  of 
breadth  to  length  being  required  for  speed.  An  absolute 
length  is  required  for  entrance  and  run,  but,  these  being 
formed  in  accordance  with  the  wave  principle  for  any  given 


460 


ENGINEERING  DRAWING. 


speed,  the  breadth  may  have  any  proportion  to  that  which  the  uses  of  the  ship 
and  the  intentions  of  the  constructor  require. " 

"  The  wave  system  allows  us  to  give  the  vessel  as  much  length  as  we  please. 
It  is  by  this  means  that  we  can  give  to  a  vessel  of  the  wave  form  the  capacity  we 
may  require,  but  which  the  ends  may  not  admit.  Thus,  the  Great  Eastern, 
which  is  a  pure  example  of  the  wave  form,  has  an  entrance  or  fore  body  of  330', 
a  run  or  after  body  of  220',  and  a  middle  body  of  120',  which  was  made  of  this 
length  merely  to  obtain  the  capacity  required.  The  lengths  of  the  fore  and 
after  body  are  indicated  by  the  required  speed,  and  if  the  beam  is  fixed,  it  is 
only  by  means  of  a  due  length  of  middle  body  that  the  required  capacity, 
stability,  and  such  other  qualities  are  to  be  given  as  will  make  a  ship,  as  a  whole, 
suit  its  use." 


FIG.  1028. 


FIG.  1029. 


Length  of  entrance  of  a  vessel  for  a  10-mile  speed  should  be  42  feet,  of  run 
30  feet  ;  for  a  20-mile  speed,  168  and  120  feet ;  that  is,  the  lengths  increase  as 
the  squares  of  the  speed. 

Under  Isometrical  Drawing  are  given  illustrations  of  vessels  constructed  on 
wave-lines. 


AKCHITECTUKAL  CHAWING. 

IT  is  the  duty  of  an  architect  to  design  a  building  to  be  suitable  and  con- 
venient for  the  purposes  for  which  it  is  intended  ;  to  select  and  dispose  of  the 
materials  of  which  it  is  composed  to  withstand  securely  and  permanently  the 
stresses  and  wear  to  which  they  may  be  subjected  ;  to  arrange  the  parts  to  pro- 
duce the  artistic  effects  consistent  with  the  use  of  the  building  and  its  location, 
and  to  apply  such  appropriate  ornament  as  may  express  the  purpose  and  har- 
monize with  the  construction. 

In  domestic  architecture,  by  far  the  most  extensive  branch  of  the  profession, 
most  persons  can  give  some  idea  of  the  kind  of  building  which  they  wish  to 
have  constructed,  and  perhaps  express  by  line  the  general  arrangement  of 
rooms  ;  but  it  is  left  to  the  architect  to  settle  the  style  of  building  appropriate 
to  the  position,  to  adapt  the  dimensions  and  positions  of  rooms  and  passages 
to  the  requirements,  to  determine  the  thickness  of  walls  and  partitions,  and 
arrange  for  drainage,  heating,  and  ventilating.  The  graphical  representation 
is  left  to  the  draughtsman,  and  his  assistance  is  the  more  valuable  if  he  is  not 
only  conversant  with  practical  details,  but  understands  the  best  proportions  of 
parts;  the  necessities  of  construction,  and  the  requirements  of  building  laws. 

The  draughtsman  usually  commences  his  education  with  the  copying  of 
drawings.  Such  are  furnished  him. 

For  this  purpose,  in  Figs.  1030  to  1034,  inclusive,  are  given  plans  and  eleva- 
tions of  a  simple  house,  which  contain  representations  sufficient  for  the  informa- 
tion of  the  owner,  and  for  the  purposes  of  estimate  of  cost,  if  accompanied  with 
full  specifications.  The  size  of  our  page  has  compelled  the  titles  to  be  put  within 
the  body  of  the  drawings ;  after  copying,  place  them  outside,  and  give  good 
margin.  On  Fig.  1034  the  section  and  end-elevation  are  given  together.  This 
is  also  for  economy  of  space,  but  should  be  copied  by  the  draughtsman  in  two 
distinct  drawings,  each  of  the  full  width  of  the  building. 

Instead  of  hatching,  it  is  usual  to  give  the  walls  a  shade  of  color  or  black, 
or  in  full  black  often,  as  ••••••.  the  black  representing  the  solid  wall, 

and  the  inner  line  that  of  the  plastering. 

Details  of  Construction. — The  necessities  of  a  suitable  foundation  for  every 
structure  have  been  treated  of  (page  362),  and  that  a  good  foundation  may  be 
secured  in  an  uniformly  yielding  earth,  as  on  a  rigid  rock.  For  the  extent  or 
width  of  base,  the  draughtsman,  if  there  are  practical  examples  in  the  vicinity 
of  the  proposed  structure,  will  conform  to  the  teachings  of  practice,  and  to  the 
building  laws,  if  there  are  any  in  force.  In  general,  for  small  buildings,  cellar- 


462 


ARCHITECTURAL  DRAWING. 


J 


ARCHITECTURAL  DRAWING. 


463 


464 


ARCHITECTURAL  DRAWING, 


LJ 

LJ 

ARCHITECTURAL 


30 


466 


ARCHITECTURAL  DRAWING. 


END  ELEVATION 


SECTION. 


SCALE  :   4'  =  1  inch. 


FIG.  1034. 


ARCHITECTURAL   DRAWING. 


467 


walls,  if  of  stone  laid  in  mortar,  should  not  be  less  than  18"  thick  ;  if  of  brick, 
16",  and  the  base  6"  to  12"  wider.  For  walls  above  the  cellar,  it  will  be  found 
difficult  to  lay  stone  walls  in  mortar,  with  fair  bond  and  face,  less  than  16* 
thick.  Brick  walls  may  be  as  thin  as  8"  for  exteriors,  and  for  partitions  4". 
Brick  walls  are  usually  bonded  by  heading-courses  every  fifth 
to  seventh  course.  Where  the  outside  course  is  pressed  or  face 
brick,  these  are  laid  on  stretchers,  and  the  bond  with  the  back- 
ing may  be  thin  strap-iron,  laid  in  the  joints,  or,  by  cutting 
off  the  interior  corners  of  the  face-course,  say  every  fifth 
course,  and  laying  common  brick  diagonally  of  the  wall  rest- 
ing in  this  clipped  corner  (Fig.  1035).  The  face  of  buildings 
is  often  built  of  thin  ashlar,  which  is  secured  with  iron  an- 
chors to  the  brick  backing. 

In  most  large  cities  there  are  building  acts  in  force,  defin- 
ing thickness  of  walls  and  foundations,  to  which  all  construc- 
tions within  their  limits  must  conform.  Extracts  from  the 
New  York  law  may  be  found  in  the  Appendix. 

Openings  in  masonry-walls  are  covered  by  lintels  or  arches, 
or  both.  It  is  usual  to  place  a  stone  or  cast-iron  lintel  in  the  exterior  face 
over  openings  for  doors  and  windows,  with  a  wooden  lintel  inside  (Fig.  1036), 
and  a  relieving  arch  above.  For  larger  openings,  brick  arches  are  turned  in 
cast-iron  skew-backs,  of  which  the  thrust  is  resisted  by  a  tie-bolt  (Fig.  1037), 
or  cast-iron  lintels,  box,  or  j4,  or  roller  I-beams.  But  it  is  to  be  observed  that, 
when  the  cement  is  set,  there  is  little  or  no  thrust  from  the  arch.  The  whole 
dead  work,  or  masonry  without 
an  opening,  forms  a  monolithic 


FIG.  1035. 


FIG.  1036. 


FIG.  1037. 


beam,  and,  if  there  is  depth  enough  of  this,  the  arch  is  of  no  account.  It  is 
the  custom  in  the  north  of  Italy  to  construct  flat  lintels  of  brick,  of  consider- 
able span,  depending  entirely  on  the  mortar  for  strength. 

To  distribute  the  weight  over  the  foundation  or  walls,  it  is  very  common  to 
turn  inverted  arches  beneath  openings. 

In  old  houses,  it  was  not  unusual  to  make  the  exterior  arches  of  an  opening 
flat  or  rectangular  in  outline,  with  the  joints  radial.  This  is  now  relegated  to 
ornamental  construction. 

Concrete  Walls. — It  is  common  in  many  places  where  brick  and  stone  are 
expensive  and  gravel  is  abundant  to  make  walls  of  concrete,  in  proportions  of  one 
of  cement  to  five  to  seven  of  gravel.  The  space  requisite  for  the  wall  is  inclosed 
with  plank,  and  is  filled  in  with  concrete,  well  rammed.  Figs.  1038  and  1039 
are  plans  of  concrete  walls  with  inclosing  plank,  and  Fig.  1040  an  elevation. 


468 


ARCHITECTURAL  DRAWING. 


The  planks  are  held  by  bolts  passing  through  wall  and  plank, 
all  of  which  are  removed  after  the  wall  is  set,  and  the  bolt- 
holes  are  then  filled  with  cement.  The  thickness  of  walls 
should  be  a  little  in  excess  of  those  of  brick. 

Wooden  walls  are  framed.     Fig.  1041  represents  the  frame 


FIG.  1038. 


FIG.  1039. 


FIG.  1040. 


of  the  side  of  a  wooden  house,  in  which  A  A  are  the  posts,  B  the  plate,  C  C 
girts  or  interties,  D  D  braces,  E  sill,  F  window-posts  or  studs,  G  G  studs. 


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FIG.  1041. 


FIG.  1042. 


The  studs  at  all  door-openings  should  be  set  at  least  2*  wider,  and  3"  higher 
than  the  size  of  the  finished  opening.  It  is  not  unusual  to  have  double  studs 
(2"  X  4ff)  to  inclose  these  openings  (Fig.  1042).  This  leaves  the  doorway  more 
or  less  independent  of  the  partition. 

Usual  dimensions  of  timber  for  frame  of  common  dwelling-houses  :  sills 
6"  X  8",  posts  4"  X  8",  studs  2"  X  4"  or  3"  X  4",  girts  6"  X  the  depth  of  floor- 
joists,  plates  4"  X  6" ;  the  floor-joists  (J,  Fig.  1043)  are  notched  into  the  girts. 
The  posts  and  studs  are  tenoned  into  the  sills  and  girts.  Fig.  1045  represents 


ARCHITECTURAL  DRAWING. 


469 


a  tenon,  I  c,  in  side  and  end  elevation,  and  mortice,  a ;  the  portions  of  the  end 
of  the  stud  resting  on  the  beam  are  called  the  shoulders  of  the  tenon.  In  the 
balloon-frame  the  girts  are  omitted  ;  the  studs  are  of  the  same  length  as  the 
posts,  and  the  floor-joists  are  supported  by  a  board,  a,  3"  or  4"  X  I",  let  into 

the  studs  (Fig.  1044),  and  firmly 
nailed  ;  the  joists  are  also  nailed 
strongly  to  the  studs. 

The  frame  is  covered  with  boards 
usually  1"  thick,  laid  either  horizon- 


FIG.  1043. 


FIG.  1044. 


I 

QD 


FIG.  1045. 


tally  or  diagonally,  and  nailed  strong- 
ly to  the  posts  or  studs.  Fig.  1046 
is  the  elevation  of  the  end  frame  of 
a  house,  showing  by  breaks  the  diag- 
onal cover  of  boards  and  the  inner 
lathing.  The  lower  story  is  sheathed 
or  ceiled  with  narrow  boards,  the  up- 
per shingled.  With  balloon  frames, 
the  bracing  depends  largely  on  the 
diagonal  boarding. 

Partitions  are  usually  simply 
studs  set  at  intervals  of  12  or  16 
inches,  these  spaces  being  adapted  to 
the  length  of  the  lath  (48  inches). 
The  sizes  of  the  studs  are  generally 
2X4,  3X5,  or  3x6  inches,  ac- 
cording to  the  height  of  the  parti- 
tion ;  for  very  high  partitions,  greater 
depth  may  be  required  for  the  studs, 
but  three  inches  will  be  sufficient 
width. 

Partitions  are  usually  cut  in  be- 
tween sills  placed  on  the  floor-beams 
(Fig.  1047),  and  similar  caps  above, 
beneath  the  beams.  Where  parti- 
tions of  the  second  story  are  directly 
above  those  on  the  first  story  it  is 
better  to  foot  the  studs  on  the  caps 
of  the  latter,  and  not  on  the  beams 
(Fig.  1048).  Where  there  are  double 
floors,  the  sills  are  placed  on  the  bot- 


470 


ARCHITECTURAL  DRAWING. 


torn  floor,  or  on  the  floor  without  a  sill.     It  may  be  important  that  the  parti- 
tions should  be  self-sustaining.      This   is  effected  by  simple  bridging,  well 


FIG.  1049 


IG.  1048. 


FIG.  1050. 


nailed  to  the  studs,  as  shown  in  Fig.  1049,  or  by  herring-bone  bridge,  as 
shown  in  plan  of  floor  (Fig.  1051),  or  by  a  system  of  trussing,  as  in  Fig.  1050. 
This  method  of  truss- 
ing must  vary  with 
the  position  of  open- 
ing. The  foot  of  the 
braces  should  rest 
on  a  positive  sup- 
port. 

The  bridging 
should  be  accurate- 
ly cut  and  firnily 
nailed.  Bridging 
distributes  the 
weight  of  the  partition,  but  trussing  concentrates  it  at  the  ends  of  the  braces. 

Flooring. — The  timbers  which  support  the  flooring-boards  and  ceiling  of  a 
room  are  called  the  naked  flooring. 

The  simplest  form  of  flooring,  and  the  one  usually  adopted  in  the  construc- 
tion of  city  houses  and  stores,  is  represented  in  plan  and  section  (Fig.  1051). 
It  consists  of  a  single  series  of  beams  or  deep  joists,  reaching  from  wall  to  wall. 
As  a  lateral  brace  between  each  set  of  beams  a  system  of  bridging  is  adopted, 
of  which  the  best  is  the  herring-bone  bridging,  formed  of  short  pieces  of  joists 
about  2x3,  crossing  each  other,  and  nailed  securely  to  the  tops  and  bottoms  of 
the  several  beams,  represented  by  a  and  b  ;  and  wherever  a  flue  occurs,  or  a 
stairway  or  well-hole  prevents  one  or  more  joists  from  resting  on  the  wall,  a 
header,  II,  is  framed  across  the  space  into  the  outer  beams  or  trimmer-beams 
T  T,  and  the  beams  cut  off  or  tail-beams  are  framed  into  the  trimmer. 

Whenever  the  distances  between  the  walls  exceed  the  length  that  can  safely 
be  given  to  joists  in  one  piece,  an  intermediate  beam  or  girder,  running  longi- 
tudinally, is  introduced,  on  which  the  joist  may  be  set  (Fig.  1052),  notched  on 
(Fig.  1053),  or  boxed  in  (Fig.  1054),  or  both  boxed  and  notched.  They  may 
also  be  framed  in  with  tenon  and  mortice ;  the  best  form  is  the  tusk-tenon 


AKCHITECTURAL  DRAWING. 


471 


FIG.  1051. 


(Fig.  1055).  Flooring  is  still  further  varied,  by  framing  with  girders  longi- 
tudinally ;  beams  crosswise,  and  framed  into  or  resting  on  the  girders ;  and 
joists  framed  into  the  beams,  running  the  same  direction  as  the  girders.  It  is 


\ 


FIG.  1052. 


Fm.  1053. 


Fia.  1054. 


FIG.  1055. 


evident,  when  the  joists  are  not  flush  or  level  with  the  bottom  of  the  beams  or 
girders,  either  that  in  the  finish  the  beams  will  show,  or  that  ceiling-  joists  or 
furrings  will  have  to  be  introduced. 

On  the  Size  of  Joists.  —  The  following  dimensions  may  be  considered  as  safe 
sizes  for  ordinary  constructions,  the  distances  from  center  to  center  being  one 
foot. 

Joists  in  floors,  clear  bearing  — 

Exceeding    7  feet,  and  not  exceeding  10  feet,  to  be  not  less  than     6X2  inches. 


10 
12 

14 
16 

18 
20 


12  " 

14  " 

16  " 

18  " 

20  " 

22  " 

24  " 


6X3 

7X3 

9X3 

9X3 

10  X  3 

11X3 

12  X  3 


It  is  to  be  observed  that  lumber  is  seldom  sawed  to  dimensions  of  fractions 
of  an  inch. 


472 


ARCHITECTURAL  DRAWING. 


Trimmer-beams  and  headers  should  be  of  greater  width  than  the  other 
beams,  depending  on  the  distance  of  the  headers  from  the  wall,  and  the  num- 
ber of  tail-beams  framed  into  it.     The  New  York  Building  Act  requires  that 
all  headers   should   be  hung  in   stirrup-irons  (Fig. 
1056),  and  not  framed  in. 

Floors. — In  New  York  it  is  usual  to  lay  single 
floors  of  tongued  and  grooved  boards  directly  on  the 
beams,  but  in  the  Eastern  States  double  floors  are 
more  common.  The  first  floor  consists  of  an  inferior 
quality  of  boards,  unmatched,  laid  during  the  prog- 
ress of  the  work  as  a  sort  of  staging  for  the  carpenter 

and  mason,  and,  in  finishing,  a  second  course  is  laid  on  them  of  better  material, 
generally  tongued  and  grooved,  but  sometimes  only  jointed.  Ceilings  should 
always  be  furred,  and  the  laths  be  nailed  to  the  strips.  Furring-strips  usually 

are  of  inch  board,  2"  wide,  and 

r- , „___..  /' ^_^_^_^_  12"  from  center  to  center,  nailed 

across  from  joist  to  joist. 

Fig.  1057  represents  a  section 


FIG.  1056. 


of  a  mill-floor.      The  girders  or 
beams,  generally  in  pairs,  with  a 

space  of  about  an  inch  between  them,  are  placed  at  a  distance  of  from  seven 
to  nine  feet  from  center  to  center,  and  are  of  from  twelve  to  sixteen  inches  in 
depth.  On  these,  a  tongued  and  grooved  plank  floor  of  from  3"  to  4"  thick 
is  laid. 

Fig.  1058  is  the  section  of  a  beam  and  mill-floor  now  adopted  as  a  fire-retard- 
ing construction,  and  considered  superior  to  iron  beams  and  brick  arches.  It 
consists  of  the  usual 
beam  and  plank  floor  ; 
both  plastered  on  the 
under  side  and  on  the 
lateral  surfaces.  The 
lathing  consists  of  wire 
cloth  stapled  through 
furring  strips  f *  to  y, 
and  then  the  usual 
three-coat  plaster.  In 
addition,  it  is  common 
to  lay  roofing-felt  on 
the  upper  surface  of  FIG.  1058. 

the  plank,  with  1"  to 

1%"  of  cement  mortar  ;  with  the  usual  floor  on  the  top  of  this,  the  floor  being 
nailed  to  strips  attached  to  the  plank,  and  serving  as  guides  to  surface  the 
cement  mortar.  Various  methods  are  given  (page  238)  of  trussing  beams 
when  the  spans  or  loads  are  in  excess  of  the  strength  of  lumber  of  the  usual 
dimensions. 

Joinings. — As  timber  can  not  always  be  obtained  of  sufficient  lengths  for 
the  different  portions  of  a  frame,  or  to  tie  the  walls  of  a  building,  it  is  often 


AECHITECTURAL  DRAWING. 


473 


necessary  to  unite  two  or  more  pieces  together  by  the  ends,  called  scarfing  or 
lapping.  Fig.  1059  is  a  most  common  means  of  lapping  or  halving  employed 
when  there  is  not  much  longitudinal  stress,  and  when  a  post  is  to  be  placed 
beneath  the  lower  joint. 

V 


FIG.  1059. 


FIG.  1060. 


Fig.  1060  is  a  long  scarf,  in  which  the  parts  are  bolted  through  and  strapped, 
suitable  for  tie-beams.  Joints  (Figs.  1061,  1062,  and  1063)  are  also  often  made 
by  abutting  the  pieces  together,  and  bolting  splicing -pieces  on  each  side  ;  still 
further  security  is  given  by  cutting  grooves  in  both  timbers  and  pieces,  and 
driving  in  keys,  Tc  k. 


',' 

','                                                            > 

!      It                     >. 

jl 

i         1                  J 

_j 


I I 


FIG.  1061. 


0  O 

o  o 


o  o 

o 


O       o 


FIG.  1062. 


EH 

s 

FIG.  1063. 


Floor-beams  in  a  building  acting  as  ties  are  usually  strapped,  or  anchored 
together  by  iron  bars,  spiked  to  the  top  or  bottom  of  the  beams,  often  sunk 
into  the  beam. 


FIG.  1064. 


FIG.  1065. 


FIG.  1066. 


Figs.  1064,  1065,  and  1066  are  common  forms  of  anchors.     The  first  two 
for  connecting  beams,  the  last  for  beams  and  walls.     In  warehouses,  it  is  usual 


ARCHITECTURAL  DRAWING. 

to  carry  the  anchors  entirely  through  the  wall,  with  a  washer  and  nut  outside. 
The  beams  are  often  joint- bolted  together  like  stair-rails. 

Fire-resisting  Floors. — Flames  spread  through  buildings  by  means  of  the 
spaces  left  between  floors  and  ceilings,  and  between  walls  and  f urrings  and  hol- 
lows in  partitions,  which  act  as  flues.  But  when  the  wooden  beams  and  plank 
floors  are  protected  beneath  by  wiie  netting  and  plaster  there  are  no  air-spaces 
for  circulation,  and  sufficient  stay  is  made  in  the  progress  of  the  flames  to  ad- 
mit of  the  application  of  means  for  extinguishment.  And  if  the  beams  are 
placed  close  together,  and  the  joints  filled  with  cement,  there  is  still  greater 
security.  Experiments  were  made  in  Paris  on  asphalt  floors  laid  on  plank,  and 
they  resisted  for  a  very  long  time  the  spread  of  flames,  both  when  fires  were 
kindled  beneath  the  floors,  and  directly  on  top  of  the  asphalt.  In  the  latter  case, 
a  thin  layer  carbonized,  and  afforded  a  good  fire-proof  material. 

Iron  beams  and  brick  arches,  as  in  Fig.  1067,  are  the  usual  form  of  fire- 
proof floors,  but  when  efficient  protection  against  fire  is  desired  the  bottom 
flange    must    be 
covered    entirely 
with    some    fire- 
proof material,  to 

prevent    contact  FIG. 

with    flame    and 

excess  of  local  heat,  tending  to  warp  and  twist  the  beams.  Iron  often  be- 
comes necessary  for  spans  greater  than  can  be  met  by  wooden  beams,  and  they 
should  be  protected  by  some  fire-proof  covering. 

Fig.  1068  represents  a  section  of  one  of  the  French  systems  of  fire-proof 
floors.  It  consists  of  I-girders,  placed  at  a  distance  of  one  metre  (39 '38  inches) 

from  center  to  center,  slight- 
ly  cambered  or  curved  up- 
ward in  the  center,  the  depth 
of  the  girders  to  depend 
upon  the  span.  Stirrups  of 
cast-iron  are  slid  upon  the 
FIG.  1068.  girders,  into  which  the  ends 

of  flat  iron  joists,  set  edge- 
ways, pass  and  are  secured  by  pins  ;  the  ends  of  the  joists  take  a  bearing  also 
on  the  bottom  flanges  of  the  girders.  The  joists  are  placed  at  a  distance  of 
one  metre  from  center  to  center.  Upon  the  joists  rest  rods  of  square  iron, 
which  in  this  way  form  a  grillage  for  the  support  of  a  species  of  rough-cast 
and  the  ceiling.  By  this  and  other  very  similar  systems,  the  French  have  suc- 
ceeded in  reducing  the  cost  of  such  floors  to  that  of  wooden  ones. 

The  dimensions  of  beams  and  girders  for  the  above  constructions  can  readily 
be  determined  from  rules  given  (page  233).  The  brick  arches  (Fig.  1067)  are 
usually  in  single  ring  or  rolock  courses,  and  beams  spaced  from  3'  to  6'  cen- 
ters. Strips  of  plank  are  fastened  on  the  top  or  at  the  side  of  the  beam  to 
receive  the  floor,  and  the  spandrels  leveled  up  with  concrete. 

Floors  constructed  of  concrete,  in  plain  cylindrical  or  groined  arches  (Fig. 
1069),  are  cheap  and  efficient  constructions.  One  of  the  warehouses  of  the 


ARCHITECTURAL  DRAWING. 


475 


FIG.  1069. 


publishers  is  covered  by  arches  of  this  last  form.     Posts  of  brick,  2  feet  square, 

13  feet  centers,  arches  arcs  of  circles,  depth  of  concrete  at  spring  21",  at  key 

9"  to  a  level  floor,  supporting  presses. 

In  Italy  ceilings  are  made  in  single  courses  of 

brick,  and  groined,  laid  without  centers,  the  arcs 

being  described  on  the  side-walls,  and  the  bricks 

laid  to  a  line  in  plaster.     The  spandrels  may  be  lev- 
eled up  with  concrete,  when  rooms  above  are  to  be 

occupied,  but  often  there  is  only  the  brick  arch 

forming  the  ceiling  of  the  principal  rooms,  with  a 

light  wooden  roof  above. 

Figs.  1070  to  1073  are  illustrations  of  Koman 

constructions  in  masonry,  from  "  Dictionnaire  Raisonne  de  1' Architecture," 

par  M.  Viollet  Le  Due. 

Fig.  1070  is  a  perspective  view  of  a  cylindrical  arch  in  process  of  construc- 
tion. The  cen- 
ters A  and  lag- 
ging B  are  quite 
light,  as  the  full 
load  of  the  arch 
is  never  borne  by 
them.  On  the 
lagging,  B,  a  cov- 
er of  flat  tile,  C, 
is  laid  in  cement, 
and  above  ribs, 
D  D,  and  girts, 
E  E,  in  brick  ma- 
sonry, shown  on 
a  larger  scale  in 
Fig.  1071,  with 
the  plank  P  used 
for  the  support  of 
the  girt  bricks  E, 
which  is  removed 
after  the  mortar 
is  set.  The  pan- 
els are  now  filled 
with  concrete. 

Fig.  1072  rep- 
resents rib  and 
portionsofgirtsof 
a  groin  shown  in 
plan,  Fig.  1073,  ef 
g  h  being  that  of 
the  rib;  K,  a  tim- 
ber of  the  center. 


FIG.  1070. 


\ 


476 


ARCHITECTURAL   DRAWING. 


A  similar  construction  also  obtained 
for  domes,  the  girts  being  of  the  same 
width  as  the  ribs,  and  sunk  panels  formed 
by  furring  up  on  the  wooden  lagging  of 
the  centers. 

Fig.  1074  is  a  perspective  of  a  dome, 
in  which  the  brick  skeleton,  ribs,  and 
girts  are  curved,  with  panels,  B  B,  of  con- 
crete. 

Doors. — In  stud-partitions,  the  open- 
ings for  doors  are  framed  as  in  Fig.  1043, 
the  door-frame  being  independent  of  the 
studs. 

Fig.  1075  represents  the  elevation  and 
Fig.  1076  the  horizontal  section  of  a 
common  inside-door.  A  A  are  the  stiles, 
B,  C,  H,  D,  the  bottom,  lock,  parting,  and 
top  rail,  E  the  panels,  and  F  the  muntin  ; 
the  combination  of  moldings  and  offsets 
around  the  door,  G,  is  called  the  archi- 
trave ;  in  the  sec- 
tion, a  a  are  the 
partition-studs,  b  b 
the  door-jambs. 

Fig.  1077  rep- 
resents the  forms 
of  the  parts  of  a 
door,  and  the  way 
in  which  they  are 
put  together. 
When  the  tenons 
are  to  be  slipped 
into  the  mortises, 
they  are  covered 
with  glue,  and, 
after  being  closed 
up,  keys  are  driv- 
en in. 

With  regard  to 
the  proportions  of 
internal  doors, 
they  should  de- 
pend in  some  de- 
gree on  the  size  of 
the  apartments  ; 
in  a  small  room  a 
large  door  always 


FIG.  1073. 


FIG.  1072. 


FIG.  1074. 


ARCHITECTURAL  DRAWING. 


477 


gives  it  a  diminutive  appearance,  but  doors  leading  from  the  same  room  or 
passage,  which  are  brought  into  the  same  view,  should  be  of  uniform  height. 
The  smaller  doors  which  are  found  on  sale  are  2  feet  4  inches  X  6  feet ;  for 
water-closets,  or  very  small  pantries,  they  are  sometimes  made  as  narrow  as  15 
inches,  but  any  less  height  than  6  feet  will  not  afford  requisite  head-room ;. 
2  feet  9  inches  X  7  feet,  3  feet  X  7  feet  6  inches,  or  3  feet  6  inches  X  8  feet, 

are  well-proportioned,  six-pan- 
eled doors.  But  the  apparent 
proportions  of  a  door  may  be 


FIG.  1075. 


12 


2 

FIG.  lore. 


FIG.  1077. 


varied  by  the  omission  of  the  parting-rail,  making  the  door  four-paneled, 
or  narrowed  still  more  by  the  omission  of  the  lock-rail,  making  a  two-pan- 
eled door.  Sometimes  the  muntin  is  omitted,  making  but  one  panel ;  but 
this,  of  course,  will  not  add  to  the  appearance  of  width,  but  the  reverse. 
Wide  panels  are  objectionable,  as  they  are  apt  to  shrink  from  the  moldings  and 
crack.  The  moldings  are  generally  planted  on,  and  nailed  to  the  stiles  and 
rails,  but  sometimes  formed  on  them. 

When  the  width  of  the  door  exceeds  five  feet,  it  is  generally  made  in  two 
parts,  each  part  being  hung  to  its  side  of  the  frame,  or  one  part  hung  to  the 
other,  so  as  to  fold  back  like  a  shutter  ;  or  the  parts  may  be  made  to  slide  back 
into  pockets  or  grooves  in  the  partition.  The  doors  may  be  supported  on 
wheels,  and  run  on  tracks  at  the  floor-level ;  or  the  tracks  may  be  above  the 
doors,  and  the  doors  suspended ;  or  they  may  be  supported  by  levers,  and  be 
moved  parallel  without  rollers. 


478 


ARCHITECTURAL   DRAWING. 


Figs.  1078,  1079,  and  1080  are  the  elevation,  vertical  and  horizontal  sections 
of  a  pair  of  sliding-doors.     There  are  no  knobs,  but  countersunk  pulls  to  the 


0 


FIG.  1078. 


FIG.  1079. 


FIG.  1080. 


doors,  that  they  may  be  slid  entirely  within  the  pockets,  with  a  special  handle 
in  the  locks  at  the  edges  of  the  doors  for  withdrawing  them. 


ARCHITECTURAL  DRAWING. 


479 


FEET 


Figs.  1081  and  1082  are  vertical  and  horizontal  sections  of  the 
same  doors  hung  on  butts  or  hinges. 

Figs.  1083  and  1084  are  the  elevation  and  horizontal  section 
of  an  antae-finished  outside-door,  with  the  side-lights  C  0,  and  a 
top,  fan,  or  transom  light  B.  The 


bar  A  is  called  a  transom,  and  this  term  is  applied  generally  to 
horizontal  bars  extending  across  openings,  or  even  across  rooms. 

Fig.  1085  is  the  elevation  of  an  outside  folding-door.    The  plan 
(Fig.  1086)  shows  a  vestibule,  V,  and  an  interior  door.     The  outer 
FIO.  losi.       doors  open,  as  shown  by  the  arcs,  and  fold  back  into  the  pockets 
or  recesses,  p  p,  in  the  wall.    This  is  a  very  common  form  of  doors 
for  first  class  houses  in  this  city.     The  fan-lights  are  made  semicircular,  and 
also  the  head  of  the  upper  panels  of  the  door  ;  these  panels  in  the  interior  or 
vestibule  door  are  of  glass. 

Windows  are  usually  understood  to  be  glazed  apertures.  The  sashes  may  be 
stationary,  but  for  most  positions  they  are  made  to  open  either  by  sliding  verti- 
cally, or  laterally,  or  like  doors.  The  first  is  the  common  form  of  window,  and 
the  sashes  are  generally  balanced  by  weights  ;  the  second,  except  in  a  cheap  form 
in  mechanics'  shops,  are  seldom  used ;  the  third,  often  used  for  access  to  bal- 


480 


ARCHITECTURAL  DRAWING. 


conies    or    between 
rooms,    are     called 
casements,     or 
French  windows. 
Figs.    1087  and 

1088  are  the  outside 
elevation  and  hori- 
zontal section  of  one 
half   of   a  common 
box-frame,  and  Fig. 

1089  a  vertical  sec- 
tion of  the  same  in 
a     wooden     frame 
house.     S  is  the  sill 
of   the    sash-frame, 
W    the    frame-sill, 
with  a  wash  to  dis- 
charge the  water,  B 
the  bottom   rail  of 
the    sash,     M    the 
meeting  rails,  T  the 
top  rail,  H  the  head 
of   the    sash-frame, 
and    A   the    archi- 
trave similar  to  that 
around  doors.     In- 
stead  of   two   sills, 
S    and    W,   one    is 
often  used,  and  in- 
clined to  form  the 
wash.      D     is     the 
common     outside 
blind.     In  the  sec- 
tional    plan     (Fig. 
1094),  0  0'  are  the 
window-stiles,  F  the 
pulley-stile,  w  w  the 
sash -weights,  p  the 
parting  strip,  and  D 
D  double-fold  shut- 
ters. 

Figs.  1090  and 
1091  are  the  inte- 
rior elevation  and 
vertical  section  of  a 
box -frame  window 
in  a  masonry  wall ; 


Fra.  1087. 


FEET 


FIG.  1088. 


FIG.  1089. 


ARCHITECTURAL  DRAWING. 


481 


Fig.  1092  is  an  exterior  view  of  the  same  window, 
and  Fig.  1093  a  horizontal  section. 

Unless  the  windows  begin  from,  or  nearly  from, 
the  floor,  the  point  a  (Fig.  1089)  may  be  fixed  at  a 


31 


FIG.  1090. 


FIG.  1091. 


482 


ARCHITECTURAL  DRAWING. 


height  of  about 
30  inches  above 
the  floor,  and  the 
top  of  the  win- 
dow sufficiently 
below  the  ceiling 
to  allow  space  for 
the  architrave  or 
other  finish  above 
the  window,  and 
for  the  cornice  of 
the  room,  if  there 
be  any;  a  little 
space  between 
these  adds  to  the 
effect.  For  com- 
mon windows, 
the  width  of  the 
sash  is  4  inches 
more  than  that  of 
the  glass,  and  the 
height  6  inches 
more ;  thus  the 
sash  of  a  window 

3  lights  wide  and 

4  lights  high,  of 
12"X16"    glass, 
is  3  feet  4  inches 
wide  and  5  feet 
10   inches   high. 
In    plate -glass 
windows    more 
width    is    taken 
for  the  stiles  and 
rails.     The  usual 
sizes  of  cylinder 
glass  are  7"  X  9" 
up  to  24"  X  36", 
but  single  thick 
glass  may  be  had 
up  to  40"  X  60"; 
double      thick, 
48"X62".    Plate 
glass,  polished  or 
rough,  may  be  had 
of  a  size  as  large 
as  14  X  8  feet. 


ARCHITECTURAL  DRAWING. 


483 


In  Fig.  1087  the  blind  D  is  hinged  to  the  hanging  stile,  and  folds  within 
the  opening  in  the  masonry.  The  slats  are  movable  on  pin  tenons,  and  those 
of  each  half,  upper  and  lower,  are  connected  by  a  central  bar,  so  that  they  are 
moved  together,  and  adjusted  at  any  angle  to  the  light.  In  Fig.  1093  the 
blinds  are  inside,  4-fold,  and  folding  back  into  pockets.  It  is  more  usual  to 
make  the  pockets  for  the  blinds  inclined  to  the  window,  as  in  Fig.  1094,  giv- 
ing to  the  interior  more  light,  or  ampler  space  for  curtains. 

Fig.    1095    is    the 


outside  elevation  of  a 
French  window  or  case- 
ment. 

Fig.  1096  represents 
the  sectional  elevation 


777$} 


FIG.  1095. 


FIG.  1096. 


FIG.  1094. 


of  the   same  window, 

in    broken   lines,    and 

on  a  larger  scale ;  the 

same  letters  designate 

similar  parts  as  in  Fig. 

1089.     A  transom-bar 

is  often  framed  between  the  meeting-rails,  and  in  this  case  the  upper  sash  may 

be  movable  ;  in  Fig.  1096  it  is  fixed.     An  upright,  called  a  mullion,  is  often 

introduced  in  the  center,  against  which  the  sash  shuts. 

For  use  as  doors,  the  lower  sashes  should  not  be  less  than  5  feet  6  inches 
high.  It  will  be  seen  that  in  these  forms  of  sash  the  rails  and  stiles  are  wide, 
and  that  for  the  same  aperture  the  French  window  admits  the  least  light.  The 
chief  objection  to  this  window  lies  in  the  difficulty  of  keeping  out  the  rain  at 
the  bottom  in  a  driving  storm.  To  obviate  this,  the  small  molding  d,  with 
a  drip  or  undercut,  is  nailed  to  the  bottom  rail ;  but  the  more  effectual  means 
is  the  patent  weather-strip,  the  same  as  used  on  outside  doors. 

Dormer  or  attic  windows  are  framed  and  set  as  in  an  upright  stud-partition. 

In  all  architectural  finish  moldings  are  a  necessity,  the  simpler  forms  of 
which  are  taken  from  Greek  or  Roman  examples. 

Greek  and  Roman  Moldings. — The  regular  Greek  moldings  are  eight  in 


484 


ARCHITECTURAL   DRAWING. 


number  :  the  Fillet  or  Band,  Torus,  Astragal  or  Bead,  Ovolo,  Cavetto,  Cyma 
Recta  or  Ogee,  Cyma  Reversa  or  Talon,  and  Scotia. 

The  fillet  («,  Fig.  1097)  is  a  small  rectangular  member,  on  a  flat  surface, 
whose  projection  is  usually  made  equal  to  its  height. 


FIG.  1097. 


FIG.  1098. 


\J 


FIG.  1099. 


The  torus  and  astragal  are  semicircles  in  form,  projecting  from  vertical 
diameters,  as  in  Fig.  1098.  The  astragal  is  distinguished  from  the  torus  in 
the  same  order  by  being  made  smaller.  The  torus  is  generally  employed  in  the 
bases  of  columns  ;  the  astragal,  in  both  the  base  and  capital. 

The  ovolo  is  a  member  strong  at  the  extremity,  and  intended  to  support. 
The  Roman  ovolo  consists  of  a  quadrant  or  a  less  portion  of  a  circle  (Fig.  1099). 
The  Greek  ovolo  is  elliptic. 

To  describe  the  Greek  ovolo  (Fig.  1100)  :  Draw  df  from  the  lower  end  of 
the  proposed  curve,  at  the  required  inclination ;  draw  the  vertical  g  ef  to  define 
the  projection,  the  point  e  being  the  extreme  point  of  the  curve.  Draw  e  h 
parallel  to  d /,  and  draw  the  vertical  d  h  k,  such  that  d  h  is  equal  to  h  k. 
Divide  e  li  and  ef  into  the  same  number  of  equal  parts  ;  from  d  draw  straight 
lines  to  the  points  of  division  in  ef,  and  from  k  draw  lines  through- the  divis- 
ions in  e  h  to  meet  those  others  successively.  The  intersections  so  found  are 
points  in  the  curve,  which  may  be  traced  accordingly. 

The  cavetto  is  described  like  the  Roman  ovolo — by  circular  arcs,  as  shown 
in  Figs.  1101  and  1102.  Sometimes  it  is  composed  of  two  circular  arcs  united 
(Fig.  1103)  ;  set  off  b  e,  two  thirds  of  the  projection,  draw  the  vertical  b  d  equal 
to  b  e,  and  on  d  describe  the  arc  b  i.  Join  e  d  and  produce  it  to  p  ;  draw  i  n 
perpendicular  to  e  d,  set  off  n  o  equal  to  ni,  and  draw  the  horizontal  line  op 
meeting  ep  ;  on  p  describe  the  arc  io  to  complete  the  curve. 


FIG.  1100. 


I 


FIG.  1101. 


FIG.  1102. 


FIG.  1103. 


The  ogee,  or  cyma  recta  (Fig.  1104),  is  compounded  of  a  concave  and  a  con- 
vex surface.  Join  a  and  b,  the  extremities  of  the  curve,  and  bisect  a  b  at  c  ;  on 
a,  c,  as  centers,  with  the  radius  a  c,  describe  arcs  cutting  at  d;  and  on  b,  c, 
describe  arcs  cutting  at  e.  On  d  and  e,  as  centers,  describe  the  arcs  a  c,  cb, 
composing  the  molding. 

The  cyma  reversa,  or  talon  (Fig.  1105),  is  a  compound  curve,  distinguished 
from  the  ogee  by  having  the  convex  part  uppermost. 


ARCHITECTURAL  DRAWING. 


485 


If  the  curve  be  required  to  be  made  quicker,  a  shorter  radius  than  a  c  must 
"be  employed.  The  projection  of  the  molding  n  I  (Fig.  1104)  is  usually  equal 
to  the  height  a  n. 

To  describe  the  Greek  talon:  Join  the  extreme  points  a,  ~b  (Fig.  1106) ;  bisect 
a  b  at  c,  and  on  a  c,  c  b,  describe  the  semicircles  b  d  c  and  c  a.  Draw  perpendicu- 
lars d  o,  etc.,  from  a  number  of  points  in  a  c,  c  b,  meeting  the  circumferences  ; 


FIG.  1104. 


FIG.  1106. 


I     <L 

FIG.  1107. 


and  from  the  same  points  set  off  horizontal  lines  equal  to  the  respective  perpen- 
diculars :  o  n  equal  to  o  d,  for  example.  The  curve  line  b  n  a,  traced  through 
the  ends  of  the  lines,  will  be  the  contour  of  the  molding. 

To  describe  a  scotia :  Divide  the  perpendicular  a  b  (Fig.  1107)  into  three 
equal  parts,  and  with  the  first,  a  e,  for  radius,  on  e  as  a  center,  describe  the  arc 
afh,  in  the  perpendicular  c  o  set  off  c  I  equal  a  ey  join  e  Z,  and  bisect  it  by  the 
perpendicular  o  d,  meeting  c  o  at  o,  on  the  center  o,  with  o  c  for  radius,  complete 
the  figure  by  the  arc  c  h. 

These  moldings,  and  combinations  of  them,  are  stuck  in  wood,  and  are  to 
be  purchased  in  every  variety.  Fig.  1108  represents  some  of  the  common 
forms  always  to  be  had,  and  of  suitable  sizes. 

Stairs  consist  of  the  tread  or  step  on  which  we  set  our  feet,  and  risers, 
upright  pieces  supporting  the  treads — each  tread  and  riser  forms  a  stair.     If 
the  treads  are  parallel  they  are  called  fliers  ;  if  less  at  one  end  than  the  other, 
they  are  called  winders,  f  and  w  (Fig. 
1115).     The  top  step,  or  any  interme- 
diate wide  step,  for  the  purpose  of  rest- 
ing, is  called  a  landing.     The  height 


to  &/. 


FIG.  1109. 


FIG.  1110. 


from  the  top  of  the  nearest  step  to  the  ceiling  above  is  called  the  headway. 
The  rounded  edge  of  the  step  is  called  a  nosing  (a,  Fig.  1109) ;  if  a  small  hol- 
low (b)  be  glued  in  the  angle  of  the  nosing  and  riser,  it  is  called  a  molded 


486 


ARCHITECTURAL  DRAWING. 


FIG.  1108. 


ARCHITECTURAL  DRAWING. 


487 


nosing.     The  pieces  which  support  the  ends  of  the  stairs  are  called  strings 
(Fig.  1110) ;  that  against  the  wall  the  wall-string,  the  other  the  outer  string. 

Besides  these  strings,  pieces  of  tim- 
ber are  framed  and  placed  beneath 


\  1  1 

j 

\  \ 

j 

d—^ 

1 

j 

I 

V 

^ 

_.-/  .' 

s 

FIG.  1114. 


488 


ARCHITECTURAL  DRAWING. 


the  fliers,  when  the  stairs  are  wide  (Fig.  1111),  called  carriages.  Sometimes 
the  strings,  instead  of  being  notched  out  to  receive  the  steps,  have  the  upper 
and  lower  edges  parallel,  with  grooves  cut  in  their  inner  faces  to  receive  the 
ends  of  the  steps  and  risers  (Fig.  1112).  These  are  called  housed  strings. 
Steps  and  risers  are  secured  in  the  grooves  by  wedges  covered  with  glue,  and 
driven  in. 

For  the  rough,  strong  strings  of  warehouses  the  carriages  are  made  of  plank, 
with  grooves  to  receive  plank-treads,  and  without  risers. 

Figs.  1113  and  1114  are  elevation  and  plan  of  a  straight  run  of  stairs,  both 
partly  in  section.  N  is  the  newel-post,  n  a  baluster,  li  the  hand-rail,  w  the 
well.  In  the  section  of  the  floors,  cleats  are  shown  nailed  to  the  beams ;  on 
these  short  boards  are  nailed  to  form  a  box  for  the  reception  of  mortar  for 
deafening. 

The  opening  represented  in  the  plan  (which  must  occur  between  the  outer 
strings,  if  they  are  not  perpendicular  over  each  other)  is  called  the  well  (W, 
Fig.  1115). 

The  breadth  of  stairs  in  general  use  is  from  9  to  12  inches.  In  the  best 
staircases,  the  breadth  should  never  be  less  than  11  inches,  nor  more  than  15. 
The  height  of  the  riser  should  be  the  more,  the  less  the  width  of  the  tread ; 
for  a  15-inch  tread  the  riser  should  be  5  inches  high  ;  for  12  inches,  6^  ;  for  9 


FIG.  1115. 


FIG.  1116. 


inches,  8.  In  laying  out  the  plan  of  stairs,  having  determined  the  starting- 
point,  either  at  bottom  or  top,  as  the  case  may  be,  find  exactly  the  height  of 
the  story  ;  divide  this  by  the  height  you  suppose  the  riser  should  be.  Thus 
(Fig.  1116),  if  the  height  of  the  story  and  thickness  of  floor  be  9  feet,  and  we 
suppose  the  riser  should  be  7  inches  high,  then  108  inches,  divided  by  7  =  15f. 


ARCHITECTURAL  DRAWING. 


489 


It  is  clear  that  there  must  be  an  even  number  of  steps,  either  16  or  15  ;  to 
be  near  the  supposed  height  of  the  riser,  adopt  15,  then — 
yy~  =  7f^  inches,  height  of  riser. 

For  this  particular  case,  assume  the  breadth  of  the  step  as  10  inches,  and 
the  length  at  3  feet,  a  very  usual  length,  seldom  exceeding  4  feet  in  the  best 
staircases  of  private  houses.  For  the  plan — lay  oif  the  outside  of  the  stairs,  two 
parallel  lines  3  feet  apart,  and  space  off  from  the  point  of  beginning  14  treads 
of  10  inches  each,  and  draw  the  cross-parallel  lines. 

To  construct  the  elevation,  project  the  lines  of  the  steps  in  plan,  and  divide 
the  height,  either  on  a  perpendicular  or  by  an  inclined  line,  into  the  number 
of  risers  (15),  and  draw  cross-parallels  through  these  points  ;  or  the  same 
points  may  be  determined  by  intersection  of  the  projections  of  the  plan  with 
a  single  inclined  line  drawn  along  the  nosing  of  top  and  bottom  steps.  It  is 
to  be  observed  that  the  number  of  treads  is  always  one  less  than  the  number 
of  risers,  the  reason  of  which  will  appear  by  observing  the  elevation. 

For  the  framing  plan  the  drawing  of  the  elevation  of  stairs  is  in  general 
necessary,  to  determine  the  opening  to  be  framed  in  the  upper  floor,  to  secure 
proper  headway.  Thus  (Fig.  1116),  the  distance  between  the  nearest  stair  and 
the  ceiling  at  a  should  not  be  less  than  6  feet  6  inches  ;  a  more  ample  space 
improves  the  look  of  the  stairway ;  but  if  we  are  confined  in  our  limits,  this 
will  determine  the  position  of  one  trimmer,  the  other  will  be  of  course  at  the 
top  of  the  stairs.  When  one  flight  is  placed  over  another,  the  space  required 
for  timber  and  plastering,  under  the  steps,  is  about  6  inches  for  ordinary 
stairs. 

When  the  stairs  are  circular,  or  consist  in  part  of  winders  and  fliers,  as  in 
Fig.  1115,  the  width  of  the  tread  of  the  winders  should  be  measured  on  the 


FIG.  1117. 


490 


ARCHITECTURAL  DRAWING. 


central  line.  The  construction  of  the  elevation  is  similar  to  that  of  the  straight 
run  (Fig.  1116),  by  dividing  the  space  between  the  stories  by  a  number  of  par- 
allel lines  equal  to  the  number  of  risers,  and  intersecting  the  parallels  by  pro- 
jections from  the  plan. 

The  objection  to  all  circular  ELEVATION. 

stairs  of  this  form,  or  with  a 
small  well-hole  or  central  shaft, 
is  that  there  is  too  much  differ- 
ence between  the  width  of  the 
tread,  but  a  small  portion  being 
of  a  suitable  size.  The  hand- 
somest and  easiest  stairs  are 
straight  runs,  divided  into  land- 
ings, intermediate  of  the  sto- 
ries, and  either  continuing  then 
in  the  same  line,  or  turning  at 
right  angles,  or  making  a  full 
return. 

Fig.  1117  is  the  side  eleva- 
tion of  a  stairs  with  wrought- 
iron  string  and  rail.  The  string 
is  made  of  wrought-iron  knees, 
welded  together  continuously, 
with  a  flat  bottom-bar  riveted 
across  the  lower  angle  of  the 
knees.  The  construction  is  not 
very  stiff,  and  is  usually  sup- 
ported by  an  intermediate  round 
bar-post. 

Where  posts  can  not  be  put 
in,  it  is  better  that  the  bottom 
bar  should  be  a  carriage  or  beam 
of  I  or  channel-iron,  with  knees 
or  cast-iron  angle-blocks  riveted 
on  the  top  of  the  beam. 

It  is  not  unusual  to  make 
housed  strings  of  plate-iron,  with 
angle-irons  riveted  on  to  receive 
the  treads  and  risers.  If  the 
plate-iron  be  wide  enough  to  serve 
instead  of  balusters,  it  makes  a 
very  strong  and  stiff  carriage. 

Figs.  1118  and  1119  are  the  plan  and  elevation  of  a  cast-iron  stairs,  with  a 
central  post  or  newel  (this  term  is  applied  also  to  the  first  post  of  any  stairs). 
The  newel-ring,  tread,  and  riser  of  each  step  are  cast  in  one  piece,  and  they  are 
put  together  by  placing  one  newel-ring  upon  that  below  and  bolting  the  outer 
extremity  of  the  riser  to  the  tread  below. 


FIG.  1118. 


PLAN. 


FIG.  1119. 


ARCHITECTURAL  DRAWING. 


491 


Fig.  1120  is  a  form  of  cast-iron  stairs  with  a  well  instead  of  a  newel ;  the 
step  and  riser  are  bolted  together  by  the  flanges.  It  will  be  seen  that  one 
tread  is  wider  than  the  others ;  this  is  a  landing. 


FIG.  1121. 


FIG.  1120. 

It  is  at  times  fashionable  to  make  the  newel  a  prominent  feature  in  the  hall,, 

often  occupying  valuable  space.    It  is  sufficient  that  it  be  large  and  stiff  enough 

for  a  support  to  the  hand-rail. 

The  top  of  the  hand-rail  should,  in  general,  be  about  2'  8"  to  3'  above  the 

nosing,  and  should  follow  the  general  line  of  the  steps.  The  angles  of  the  hand- 
rail should  always  be  eased  off.  A  hand-rail,  affording 
assistance  in  ascending  or  descending,  should  not  be 
wider  than  the  grasp  of  the  hand  (Fig.  1121) ;  but  where, 
for  architectural  effect,  a  more  massive  form  may  be 
necessary,  it  is  very  convenient,  and  may  be  very  orna- 
mental, to  have  a  sort  of  double  form,  that  is,  a  smaller 
one  planted  on  top  of  the  larger  (Fig.  1122). 

To  a  draughtsman  conversant  with  the  principles  of 
projection  already  given,  it  will  not  be  difficult  to  draw 
in  the  hand-rail  of  stairs,  or  to  lay  off  the  mold  for  its 
construction.  It  will  follow  the  line  of  stair-nosing,  and 
where  there  are  changes  of  pitch  they  are  made  to  con- 
nect by  curves  tangent  to  these  pitches,  except  where  the 
landings  are  square,  and  newels  set  at  the  head  of  the 
landings,  the  rail  is  made  to  bolt  into  the  newel.  At  the 
bottom  the  rail  is  curved  to  the  horizontal,  when  it 

comes  into  or  upon  top  of  the  newel. 

Balusters  are  of  great  variety — usually  turned  forms — attached  to  the  treads 

by  dovetails,  covered  with  the  returned  nosing,  or  with  pin-ends  and  holes  in 


FIG.  1122. 


492 


ARCHITECTURAL  DRAWING. 


FIG.  1123. 


treads  and  under  side  of  caps.  Sometimes  (especially  in  iron- 
work) the  baluster  is  set  in  a  bracket  from  the  face  of  the 
string,  as  in  Fig.  1123.  These  brackets  are  often  very  orna- 
mental, and  the  balusters  may  be  cast  on  the  same  piece  with 
the  bracket. 

Fireplaces. — Fireplaces  for  wood  are  made  with  flaring 
jambs  of  the  form  shown  in  plan  (Fig.  1124)  ;  the  depth  from 
1  foot  to  15  inches,  the  width  of  opening  in  front  from  2  feet 
6  inches  to  4  feet,  according  to  the  size  of  the  room  to  be  warmed  ;  height  2 
feet  3  inches  to  2  feet  9  inches,  the  width  of  back  about  8  inches  less  than  in 
front ;  but  at  present  fireplaces  for  wood 
are  seldom  used,  stoves  and  grates  hav- 
ing superseded  the  fireplace.  The  space 
requisite  for  the  largest  grate  need  not 


FIG.  1124. 

exceed  2  feet  in  width  by  8  inches  in       _J 
depth.     The  requisite  depth  is  given  by  FIG.  1125. 

the  projection  of  the  grate,  and  the  man- 
tel-piece.    Ranges  require  from  4  feet  4  inches  to  6  feet  4  inches  wide  X  12 
inches  to  20  inches  deep ;  jambs  8  inches  to  12  inches. 

Fig.  1125  represents  the  elevation  of  a  mantel-piece  of  very  usual  propor- 
tions.    The  length  of  the  mantel  is  5  feet  5  inches,  the  width  at  base  4  feet  6 
inches,  the  height  of  opening  2  feet  7  inches,  and  width  2  feet  9  inches.     A 
portion  of  this  opening  is  covered 
by  the  iron  sides  or  architrave  of 
the  grate,  and  the  actual  open  space 
would  not  probably  exceed  18  inch- 
es in  width  by  2  feet  in  height.    In 
brick  or  stone  houses  the  flues  are 


FIG.  1126. 

formed  in  the  thickness  of  the  wall, 
but  when  distinct  they  have  an  out- 
side shell  of  a  half-brick  or  4  inches, 
and  sometimes  8"  (Fig.  1126) ;  the 
withs  or  division-walls  always  4". 


FIG.  1127. 


ARCHITECTURAL  DRAWING. 


493 


FIG.  1128. 


The  size  of  house  flues  is  usually  8"  X  8",  but  some  are  4"  X  8",  4"  X  12",  ar.d 
8"  X  12".  The  flues  of  different  fireplaces  should  be  distinct.  Those  from, 
the  lower  stories  pass  up  through  the  jambs  of  the  upper  fireplaces,  and, 
keeping  side  by  side  with  but  4-inch  brick-work  between  them,  are  topped  out 
above  the  roof,  sometimes  in  a  double  and  often  in  a  single  line  16  inches  wide 
by  a  breadth  required  by  the  number  of  flues,  as  in  Fig.  1126,  or  in  Fig.  1127. 
The  latter  is  an  illustration  of  how  far 
flues  may  be  diverted  from  a  vertical 
line,  but  it  is  to  be  observed  that  the 
construction  must  be  stable,  as  any  set- 
tling or  cracks  not  only  injures  the 
draught  of  the  chimney,  but  impairs  the 
security  of  the  building  against  fire. 
Changes  of  direction  of  flues  should 
never  be  abrupt.  The  back  of  the  fire- 
place may  be  perpendicular  through  its 
whole  height,  but  it  is  usual  to  incline 
the  upper  half  inwardly  toward  the 

room,  making  the  throat  to  the  flue  long  and  narrow.     It  is  very  common  to 
form  the  upper  3"  to  4"  of  the  inclined  back  by  an  iron  plate,  which  can  be 
turned  back  or  forward  to  increase  or  diminish  the  draught.     Fig.  1128  repre- 
sents the  arrangement  of  frame  and  brick  arch  for  the  support  of  the  hearth. 
The  chimney  is  generally  capped  with  stone,  sometimes  with  tile  or  cement 
pots.     As  an  architectural  feature,  the  chimney  is  often 
^^\^  made  to  add  considerably  to  the  effect  of  a  design. 

^\  Roofs. — Framed  roofs  have  been  illustrated  (page  410). 

/  \       City  roofs  are  usually  flat,  and  timbered  similarly  to  floors, 

r      but  not  so  strongly,  with  a  slight  pitch  to  discharge  rain- 
I       fall.     Eoofs  of  country  dwellings  are  usually  framed  like 
FIG.  1129.  stud-partitions,  with  inclined  studs  somewhat  deeper  than 

if  they  were  vertical,  depending  on  the  inclination  from 
the  vertical ;  if  flat,  depth  like  that  of  a  floor.     The  theory  of  the  construc- 
tion of  the  gambrel  or  Mansard  roof  (Fig. 
1129)  is  a  roof  with  two  kinds  of  pitch  ;  it  is 
that  of  the  polygon  of  rods,  and  self-sup- 
porting ;  but,  in  general,  they  have  central 
support  from  partitions,  and  their  outlines 
are  much  varied  by  curves  in  the  lower  raft- 
ers cut  from  plank. 

Fig.  1130  is  the  plan  of  a  roof  as  usually 
drawn,  shaded  strongly  at  the  ridges.  The 
transept  roof  is  hipped  at  A  and  B. 

Gutters  are  generally  formed  in  the  cor- 
nice (Fig.  1131) ;  sometimes  on  the  roof 

(Fig.  1132),  and  sometimes  by  raising  a  parapet  (Fig.  1133)  and  forming  a 
valley.     The  intersection  of  two  roofs  forms  a  valley. 

Fig.  1131  represents  a  form  of  gutter  very  common  to  city  buildings,  the 


FIG.  1130. 


494: 


ARCHITECTURAL  DRAWING. 


FlG.     1131. 


FIG.  1132. 


FIG.  1133. 


Toof  boarding  extending  over  the  gutter  ;  but  it  is  preferable  to  make  the  roof 
pitch  from  both  rear  and  front  to  the  center  of  the  building,  and  to  carry  the 
leader  down  in  the  interior,  where  it  may  serve  as  a  soil-pipe  for  the  water-clos- 
ets, basins,  and  baths,  affording  ventilation  in  fair  weather  and  a  scour  in  rains. 

Fig.  1134  is  a  gutter  of  a  cottage  roof. 
Fig.  1135  is  the  section  of  a  Mansard  roof,  so  called, 
showing  the  side  elevation  of  a  dormer-window,  with  the 
gutter  below  its  sill. 


FIG.  1134. 


FIG.  1135. 


It  is  to  be  observed  that  the  sheet-metal  forming  the  gutter  must  extend 
well  up  or  back  beneath  the  shingles  or  felt,  or  be  soldered  to  the  tin  of  the 
roof,  to  prevent  water  finding  its  way  into  the  interior  ;  and  at  the  sides 
flashings  of  tin  must  extend  on  the  walls  above  the  roof  and  into  the  joints  of 
the  brick. 

Plastering. — To  prevent  damp  striking  through  the  plastering  of  outer 
walls,  and  cracks  in  ceilings,  it  is  usual  to  fur  walls  and  beams  ;  that  is,  to  nail 
vertical  strips  of  wood  to  the  walls,  and  across  from  beam  to  beam.  Furring- 


ARCHITECTUKAL  DRAWIN< 


' 


495 


strips  are  from  1-J*  to  2"  wide,  and  about  J"  thick,  nailed  at  distances  of  12"  or 
16"  centers  (usually  the  former),  adapted  to  the  length  of  the  laths,  which  are 
4  feet  long,  and  about  iy  X  i"  =  spaces  between  laths  i"  to  f".  The  first  coat 
of  mortar  is  the  scratch-coat,  which  is  forced  through  the  interstices  between 


1 

K 

*^»«^ 

<= 

\... 

<±^ 

^ 

5§ 
< 

FIG.  1136. 


Fro.  1137. 


the  laths,  to  make  a  lock  to  retain  it.     This  coat  is  about  £"  tliick.     The 
next  or  brown  coat  is  about  -J-"  thick,  and  if  the  last  coat  is  a  sand-finish, 
it  will  be  less  than  -J"  thick ;  while,  if  the  last  coat  is  a  hard 
finish,  its  thickness  will  be  almost  imperceptible.     Figs.  1136 
and  1137  are  sections  of  furring  and  plastering. 

The  brown  coat  is  usually  carried  down  to  the  floor.  Over  this 
is  nailed  the  base-board,  A  (Fig.  1138),  for  the  finish  around  the 
bottom  of  the  walls  of  the  room.  Above  the  base  is  a  molding 
forming  a  part  of  the  base  ;  above  this,  there  may  be  a  molded 


FIG.  1139. 


FIG.  1140. 


FIG.  1141. 


iiliiiil 

FIG.  1138. 


rail,  B,  called  the  chair-rail,  or  surbase,  and  between  a  panel, 
termed  a  dado.     The  walls  of  stores  are  generally  ceiled  up  as 
high  as  the  surbase.     For  the  finish  of  the  angle  of  the  wall 
and  ceiling,  it  is  usual  in  the  better  rooms  to  form  a  cornice 
in  plaster.     The  cornices  are  moldings  of  varied  forms,  with 
or  without  enrichments — that  is,  plaster  ornaments.     Figs.  1139,  1140,  and 
1141  are  sections  of  cornices.     If  the  rooms  are  low,  the  cornice  should  ex- 
tend but  little  on  the  wall,  but  well  out  on  the  ceiling. 

Proportions  and  Distribution  of  Rooms  and  Passages. — Rooms  of  dwell- 
ing-houses are  to  be  proportioned  and  arranged  according  to  the  necessities  of 
position  and  use,  the  space  that  can  be  occupied,  the  financial  means  available, 
and  often  to  suit  the  peculiar  wishes  of  owners  or  occupants.  In  cities,  the 
limits  of  the  lot  restrict  the  arrangements  to  a  small  ground-space,  and  require 
an  increase  in  the  number  of  stories.  Use  has  established  certain  forms  often 
peculiar  to  different  cities,  beyond  which  there  is  little  change  ;  but  in  the 
country,  where  there  is  plenty  of  ground-space,  and  where  many  stories  are 


496  ARCHITECTURAL  DRAWING. 

usually  injurious  to  the  aesthetic  effect,  and  where  there  are  few  canons  in 
architecture  to  be  observed,  there  is  little  limit  to  the  variety  of  forms  and 
arrangements  of  country-houses. 

In  designing  a  country-house,  where  one  is  not  restricted  to  room,  it  is  often 
convenient  to  mark  out  the  rooms  of  the  desired  size  on  slips  of  paper,  accord- 
ing to  some  scale,  then  cut  them  out  and  arrange  them  in  as  convenient  an 
order  as  possible,  and  modify  the  arrangement  by  the  necessities  of  construction 
and  economy.  Thus,  the  more  the  inclosing  surface  in  proportion  to  the  in- 
cluded area,  and  the  greater  the  number  of  chimneys  and  space  used  for  pas- 
sages, the  greater  the  cost.  The  kitchen  should  be  of  convenient  access  to  the 
dining-room,  both  should  have  large  and  commodious  pantries,  and  all  rooms 
should  have  an  access  from  an  entry,  without  being  compelled  to  pass  through 
other  rooms  ;  this  is  particularly  applicable  to  the  communication  of  the  kitchen 
with  the  front  door.  Outside  doors  for  common  and  indiscriminate  access- 
should  not  open  into  important  rooms. 

As  to  the  size  of  the  different  rooms,  they  must  of  course  depend  on  the  pur- 
poses to  which  they  are  to  be  applied,  the  class  of  house,  and  the  number  of 
occupants.  The  kitchen  for  the  poorer  class  of  houses  is  also  used  as  an  eat- 
ing-room, and  should  therefore  be  of  considerable  size  to  answer  both  purposes  ; 
for  the  richer  houses,  size  is  necessary  for  the  convenience  of  the  work.  In 
New  York  city  houses  the  average  will  be  found  to  be  about  16  X  20  feet ;  for 
medium  houses  in  the  country  they  are  in  general  less,  say  12  X  16.  A  back 
kitchen,  scullery,  or  laundry,  should  be  attached  to  the  kitchen,  and  may  serve 
as  a  passage-way  out. 

The  Dining  or  Eating  Rooms. — The  width  of  dining- tables  varies  from  3  to 
5  feet  6  inches  ;  the  space  occupied  by  the  chair  and  person  sitting  at  the  table 
is  about  18  inches  ;  the  table-space,  for  comfort,  should  be  not  less  than  2  feet 
for  each  person  at  the  sides  of  the  table,  and  considerable  more  at  the  head  and 
foot ;  hence  the  space  that  will  be  necessary  for  the  family  and  its  visitors  at 
the  table  may  be  calculated.  Allow  a  further  space  of  2  feet  at  each  side  for 
passages,  and  some  3  to  5  at  the  head  for  the  extra  tables  or  chairs,  for  the 
minimum  of  space  required  ;  but,  if  possible,  do  not  confine  the  dining-room 
to  meager  limits,  unless  for  very  small  families  ;  let  not  the  parties  be  lost  in 
the  extent  of  space,  nor  let  them  appear  crowded. 

The  show-room  parlors,  if  there  are  any  intended  for  such  in  the  house, 
should  be  made  according  to  the  rules  given  below,  not  square,  but  the  length 
about  once  and  a  half  the  width  ;  if  much  longer  than  this,  break  up  the  walls 
by  transoms  or  projections.  As  to  the  particular  dimensions,  no  rules  can  be 
given  ;  they  must  depend  on  every  person's  taste  and  means  ;  20  X  16  may  be 
considered  a  fair  medium  size  for  a  regular  living-room  parlor,  not  a  drawing- 
room.  The  same  size  will  answer  very  well  for  a  sleeping-room.  The  usual 
width  of  single  beds  is  2  feet  8  inches ;  of  three-quarter,  3  feet  6  inches  ;  of 
whole,  4  feet  6  inches  ;  the  length,  6  feet  6  inches  ;  and  as  the  other  furniture 
may  be  made  to  consist  of  but  very  few  pieces,  if  adequate  means  of  ventilation 
are  provided,  it  is  easy  to  see  into  how  small  quarters  persons  may  be  thrust. 
The  bed  should  not  stand  too  near  the  fire,  nor  between  two  windows  ;  its  most 
convenient  position  is  head  against  an  interior  wall,  with  a  space  on  each  side 


ARCHITECTURAL  DRAWING.  497 

of  at  least  2  feet.  To  the  important  bedrooms  of  first-class  houses,  dressing- 
rooms  should  be  attached,  and,  if  there  is  water  and  sewer  service,  fitted  with 
set  bowls  and  baths  and  water-closets.  If  possible,  there  should  be  windows 
opening  to  the  outer  air,  but  always  with  flue-ventilation. 

Pantries. — Closets  for  crockery  should  not  be  less  than  14  inches  in  depth 
in  the  clear  ;  for  the  hanging  up  of  clothes,  not  less  than  18  inches,  and  should 
be  attached  to  every  bedroom.  For  medium  houses,  the  closets  of  large  sleep- 
ing-rooms should  be'  at  least  3  feet  wide,  with  hanging-room,  and  drawers  and 
shelves.  There  should  also  be  blanket-closets,  for  the  storing  of  blankets  and 
linen  ;  these  should  be  accessible  from  the  entries,  and  may  be  in  the  attic. 
Store-closets  should  also  be  arranged  for  groceries  and  sweetmeats. 

Passages. — Front  entries  are  usually  6  feet  wide  in  the  clear ;  common  pas- 
sage-ways, 3  feet ;  these  are  what  are  required,  but  ample  passages  give  an 
important  effect  to  the  appearance  of  the  house.  The  width  of  principal 
stairs  should  be  not  less  than  3  feet,  and  all  first-class  houses,  especially  those 
not  provided  with  water-closets  and  slop-sinks  on  the  chamber-floor,  should 
have  two  pairs  of  stairs,  a  front  and  a  back  pair  ;  the  back  stairs  need  not 
necessarily  be  over  2  feet  6  inches  in  width. 

The  Height,  of  Stories. — It  is  usual  to  make  the  height  of  all  the  rooms  on 
each  floor  equal ;  it  can  be  avoided  by  furring  down,  or  by  the  breaking  up  of 
the  stories,  by  the  introduction  of  a  mezzonine  or  intermediate  story  over  the 
smaller  rooms.  Both  remedies  are  objectionable. 

The  average  height  of  the  stories  for  common  city  dwellings  is :  Cellar,  6 
feet  6  inches  ;  common  basement,  8  to  9  feet  ;  English  basement,  9  to  10  feet ; 
principal  story,  12  to  15  feet ;  first  chamber  floor,  10  to  12  feet ;  other  chamber- 
floors,  8  to  10  feet — all  in  the  clear.  For  country-houses,  the  smaller  of  the 
dimensions  are  more  commonly  used.  Attic  stories  are  sometimes  but  a  trifle 
over  6  feet  in  height,  but  are,  of  course,  objectionable. 

Privies,  Water- Closets,  and  Out- Houses. — The  size  of  privies  must  depend 
greatly  on  the  uses  of  the  building  to  which  they  are  to  be  attached,  its  position, 
and  the  character  of  its  occupants.  Allowing  nothing  for  evaporation  and  ab- 
sorption, the  entire  space  necessary  for  the  excrementitious  deposits  of  each 
individual,  on  an  average,  will  be  about  seven  cubic  feet  for  six  months,  of 
which  three  quarters  is  fluid.  In  the  country,  vaults  are  usually  constructed  of 
dry  rubble-stone,  and  the  fluid  matters  are  expected  to  be  filtered  through  the 
earth,  the  same  as  in  cesspool-waste  ;  but  great  care  must  be  taken  that  they 
neither  vitiate  the  water-supply  nor  the  air  of  the  house.  A  brick  and  cement 
vault,  air  and  water  tight,  with  a  ventilating-pipe  into  a  hot  chimney-flue,  is 
the  best  preventive,  and  may  even  be  built  within  the  house.  In  all  other  cases 
there  should  be  free  air-space  between  the  house  and  privy.  In  the  city,  where 
there  is  adequate  water-supply  and  sewerage,  the  water-closet  should  be  adopted, 
except  in  houses  occupied  by  many  ignorant  and  irresponsible  tenants,  who 
throw  extraneous  matters  into  the  hoppers,  and  obstruct  the  sewer-pipes.  In 
these,  tight  privy-vaults,  with  trapped  sewer  connections,  and  with  all  the 
house-waste  and  roof-water  discharging  in  to  them,  are  the  easiest  kept  in  order. 
The  water-closet,  or  privy,  with  a  single  seat,  should  occupy  a  space  not  less 
than  4'x  2'  6".  The  rise  of  seat  should  be  about  17"  high  ;  and  the  hole  egg- 

32 


4:98  ARCHITECTURAL  DRAWING. 

shaped,  11"  X  8".  The  earth-closet,  when  properly  taken  care  of,  is  an  ex- 
tremely useful  appendage  to  a  country-house,  and  the  space  requisite  for  it  is 
the  same  as  that  of  a  water-closet.  It  is  the  most  common  practice  to  place 
the  water-closet  in  the  bath-room.  A  common  bath-tub  will  occupy  a  floor- 
space  of  6'  X  2',  and  18"  deep  ;  the  French  tub,  so  called,  is  much  shorter, 
often  not  over  4'  6",  but  deeper.  The  water-closet  seat  will  occupy  about  2 
feet  in  width  X  20  inches  in  depth. 

The  forms  of  modern  water  appliances,  and  the  means  to  get  rid  of  house- 
waste,  will  be  illustrated  hereafter,  under  the  heads  of  Ventilation  and 
Plumbing. 

For  Wood  or  Coal  Sheds  or  Bins. — In  estimating  the  size  of  these  accesso- 
ries, it  may  only  be  necessary  to  state  that  a  cord  of  wood  contains  128  cubic 
feet,  and  a  ton  of  coal  occupies  a  space  of  about  40  cubic  feet. 

On  the  Size  and  Proportion  of  Rooms  in  general. — "Proportion  and  or- 
nament," according  to  Ferguson,  "are  the  two  most  important  resources  at 
the  command  of  the  architect,  the  former  enabling  him  to  construct  ornament- 
ally, the  latter  to  ornament  his  construction."  A  proportion  to  be  good  must 
be  modified  by  every  varying  exigence  of  a  design  ;  it  is  of  course  impossible  to 
lay  down  any  general  rules  which  shall  hold  good  in  all  cases  ;  but  a  few  of  its 
principles  are  obvious  enough.  To  take  first  the  simplest  form  of  the  propo- 
sition, let  us  suppose  a  room  built,  which  shall  be  an  exact  cube — of  say  20  feet 
each  way — such  a  proportion  must  be  bad  and  inartistic  ;  and,  besides,  the 
height  is  too  great  for  the  other  dimensions.  As  a  general  rule,  a  square  in 
plan  is  least  pleasing.  It  is  always  better  that  one  side  should  be  longer  than 
the  other,  so  as  to  give  a  little  variety  to  the  design.  Once  and  a  half  the 
width  has  been  often  recommended,  and  with  every  increase  of  length  an  in- 
crease of  height  is  not  only  allowable,  but  indispensable.  Some  such  rule  as 
the  following  meets  most  cases  :  "  The  height  of  the  room  ought  to  be  equal  to 
half  its  width  plus  the  square  root  of  its  length  "  ;  but  if  the  height  exceed  the 
width  the  effect  is  to  make  the  room  look  narrow.  Again,  by  increasing  the 
length  we  diminish,  apparently,  the  other  two  dimensions.  This,  however,  is 
merely  speaking  of  plain  rooms  with  plain  walls  ;  it  is  evident  that  it  will  be 
impossible,  in  any  house,  to  construct  all  the  rooms  and  passages  to  conform  to 
any  one  rule  of  proportion,  nor  is  it  necessary,  for  in  many  rooms  it  would  not 
add  to  their  convenience,  which  is  often  the  most  desirable  end  ;  and,  if  re- 
quired, the  unpleasing  dimensions  may  be  counteracted  by  the  art  of  the  archi- 
tect, for  it  is  easy  to  increase  the  apparent  height  by  strongly  marked  vertical 
lines,  or  bring  it  down  by  horizontal  ones.  Thus,  if  the  walls  of  two  rooms  of 
the  same  dimensions  be  covered  with  the  same  strongly  marked  striped  paper, 
in  one  case  the  stripes  being  vertical  and  in  the  other  horizontal,  the  apparent 
dimensions  will  be  altered  very  considerably.  So  also  a  deep,  bold  cornice 
diminishes  the  apparent  height  of  a  room.  If  the  room  is  too  long  for  its  other 
dimensions,  this  can  be  remedied  by  breaks  in  the  walls,  by  the  introduction 
of  pilasters,  etc.  So  also,  as  to  the  external  dimensions  of  a  wall,  if  the  length 
is  too  great  it  is  to  be  remedied  by  projections,  or  by  breaking  up  the  lengths 
into  divisions. 

Understanding  the  general  necessities  of  a  dwelling,  the  proportions  of 


ARCHITECTURAL  DRAWING. 


500 


ARCHITECTURAL  DRAWING. 


rooms,  forms  of  construction,  and  space  to  be  occupied,  the  draughtsman  is 
prepared  to  undertake  designing,  and  for  this  purpose  cross-section  paper  will 
be  found  of  very  great  use.  Taking  the  side  of  a  small  square  as  a  unit — 
one  foot,  for  instance — he  can  readily  pencil  in  rooms  and  passages,  and  alter 
and  modify  at  pleasure. 

Figs.  1142  to  1149  are  illustrations  of  this  form  of  designing,  making  rou.sh 
sketches.  It  is  to  be  observed  that  partitions  are  to  be  as  much  as  possible 
one  over  the  other,  and  the  posts  or  walls  arranged  in  the  cellar,  for  the  sup- 
port of  these  lines  of  partitions.  For  the  sketch,  it  is  sufficient  to  make  door 
and  window  openings  3  feet,  unless  for  some  particular  purpose  bow  or  mul- 
lioned  windows  are  required.  In  arranging  the  stairs,  the  clear  space  is  roughly 
about  12  feet,  and  from  the  foot  of  the  stairs  to  the  top  H  times  the  height 
of  the  story  from  the  top  of  the  floor  to  the  top  of  the  floor,  counting  the 
square  landings  as  1  foot  each.  In  the  sketch,  the  stair-head  room  to  be  pro- 
vided for  is  that  for  the  cellar-stairs,  that  lead  from  a  small  entry  between 
the  kitchen  and  main  hall.  Chimney-breasts  may  be  sketched  as  4'  X  2'. 
When  the  sketch  is  transferred  to  drawing-paper,  the  spaces  are  then  to  be 
more  exactly  arranged  and  plotted  to  a  scale. 

Figs.  1150  to  1165  represent  plans  of  familiar  forms  of  houses,  all  drawn  to 
the  scale  of  32  feet  to  the  inch,  as  illustrations  to  the  student,  and  as  examples 
to  be  copied  on  a  larger  scale.  The  same  letters  of  reference  are  used  on  all 
the  plans,  for  rooms  intended  for  similar  purposes.  Thus,  K  K  designate 
kitchens,  cooking-rooms,  or  laundries  ;  D  D  eating-rooms  ;  S  S  sleeping-rooms  ; 
P  P  drawing-rooms,  parlors,  or  libraries ;  p p  pantries,  china  or  store  closets, 
or  clothes-presses  ;  c  c  water-closets  and  bath-rooms. 


FIG.  1150. 


FT""1'  •-"    ' 

S 

1 

S 

'rff\ 

1 

="3>: 

1  "'  M  ' 

FIG.  1151. 


FIG.  1152. 


Figs.  1150,  1151,  and  1153  are  first-story  plans  of 
square  houses,  or  of  square  outline.  Fig.  1152  is  the  sec- 
ond story  of  Fig.  1151.  This  form  of  house  has  the  great- 
est interior  accommodations  for  the  outside  cover,  and, 
although  not  picturesque  in  its  elevation,  is  a  very  con- 
venient and  economical  structure.  The  kitchen  (Fig. 
1153)  is  in  the  basement,  and  the  connection  with  the 
dining-room  is  by  a  dumb-waiter  in  the  pantry  (p).  In 
Fig.  1154  the  plan  is  the  same  as  in  Fig.  1153,  but  the 
kitchen  (k)  is  in  an  L  attached  to  the  house  ;  there  is  a  small  opening  be- 
tween the  pantry  (p')  and  kitchen,  through  which  dishes  are  passed  to  and 
from  the  dining-room. 


FIG.  1153. 


ARCHITECTURAL  DRAWING. 


501 


Fig.  1155  is  the  plan  of  a  very  small  but  convenient  floor,  of  prettier  outline 
than  the  square  ;  v  is  a  portico  or  veranda.  No  chimney  is  shown  in  the  sleep- 
ing-room S  ;  there  should  be  one  either  against  the  stairs  or  the  back  wall. 

Figs.  1156  and  1157  are  first-story  plans  of  houses  still  more  extensive. 
All  of  the  above  are  adapted  to  the  country,  dependent  on  lights  on  all  sides, 
and  ample  spaces. 


FIG.  1155. 


P         K 


K 


FIG.  1154. 


FIG.  1156. 


FIG.  1157. 


In  the  cities,  houses  are  mostly  confined  to  one  form  in  their  general  out- 
line— a  rectangle.  Figs.  1158  and  1162  may  be  taken  as  the  usual  type  of  New 
York  city  houses.  Figs.  1158,  1159,  and  1160  are  the  basement,  first  and 
second  floor  plans  of  a  three-rooms-deep,  high-stoop  house,  as  the  first  floor  is 


L 


JJ 


D 


FIG.  1158. 


FIG.  1159. 


FIG.  1160. 


FTG.  1161. 


reached  by  an  outside  flight  of  steps  about  6  feet  high.  There  is  usually  a 
cellar  beneath  the  basement,  but  in  some  cases  there  are  front  vaults,  entered 
beneath  the  steps  to  the  front  door ;  the  entrance  to  the  basement  itself  is  also 
beneath  the  steps.  The  front  room  of  the  basement  may  be  used  as  an  eating- 


502 


ARCHITECTURAL  DRAWING. 


room,  for  the  servants'  sleeping-room,  billiards,  or  library.  The  usual  dining- 
room  is  on  the  first  floor ;  a  dumb-waiter  being  placed  in  the  butler's  pantry,  p, 
for  convenience  in  transporting  dishes  to  and  from  the  kitchen.  The  objection 
to  three-rooms-deep  houses  is  that  the  central  room  is  too  dark,  being  lighted 
by  sash  folding-doors  between  that  and  the  front  or  rear  rooms,  or  both.  Fig. 
1161  is  a  modification  to  avoid  this  objection,  the  dining-room,  or  tea-room,  as 
it  is  generally  called,  being  built  as  an  L,  so  that  there  is  at  least  one  window 
in  the  central  room  opening  directly  out-doors.  This  was  an  old  fashion  here, 
and  has  lately  been  revived. 

Figs.  1162  to  1165  are  plans  of  the  several  floors  of  an  English  basement- 
house,  so  called,  distinguished  from  the  former  in  that  the  principal  floor  is  up 
one  flight  of  stairs.  The  first  story  or  basement  is  but  one  or  two  steps  above 
the  street,  and  contains  the  dining-room,  with  its  butler's  pantry  and  dumb- 


FIG.  1162. 


FIG.  1163. 


FIG.  1164. 


S 


FIG.  1165. 


waiter,  a  small  sitting-room,  with,  in  some  cases,  a  small  bedroom  in  the  space 
in  the  rear  of  it.  The  kitchen  is  situated  beneath  the  dining-room,  in  the  sub- 
basement.  The  grade  of  the  yard  is  in  general  some  few  steps  above  the  floor 
of  the  kitchen.  Vaults  for  coal  and  provisions  are  excavated  either  beneath 
the  pavement  in  front  or  beneath  the  yard.  The  advantages  of  this  form  of 
house  are  the  small  reception-room  on  the  first  floor,  which  in  small  families 
and  in  the  winter  months  is  the  most  frequently  occupied  as  a  sitting-room  of 
any  in  the  house  ;  the  spaciousness  of  its  dining-room  and  parlors  in  propor- 
tion to  the  width  of  the  house,  which  is  often  but  16  feet  8  inches  in  width,  or 
three  houses  to  two  lots,  and  not  unfrequently  of  even  a  less  width.  The  ob- 
jections to  the  house  are  the  stairs,  which  it  is  necessary  to  traverse  in  passing 
from  the  dining-rooms  or  kitchen  to  the  sleeping-rooms,  but  this  objection 
would,  of  course,  lie  against  any  house  of  narrow  dimensions,  where  floor-space 
is  supplied  by  height. 

In  New  York,  outside  access  to  the  kitchen  is  from  the  front,  as  there  is  no 
back  street  or  alley.  In  Philadelphia,  where  the  lots  are  deeper,  and  there  is 
a  street  in  the  rear,  the  kitchen  is  usually  in  a  rear  L,  on  the  level  of  the  first 
floor,  with  the  dining-room  above  it  on  a  mezzonine  or  half-story  between  the 
first  and  second  floors. 


ARCHITECTURAL  DRAWING. 


503 


Figs.  1166  to  1171  are  plans  and  elevations  of  a  country-house  in  the  Flem- 
ish or  Queen  Anne  style. 


PLAN  OF  FIRST  FLOOE. 
B 


504 


ARCHITECTURAL  DRAWING. 


PLAN  OF  SECOND  FLOOK. 


FIG.  1167. 


ARCHITECTURAL  DRAWING. 


505 


FRAMING-PLAN  OF  FIEST  FLOOR. 


506 


ARCHITECTURAL  DRAWING. 


ARCHITECTURAL  DRAWING. 
ELEVATION  OF  CHIMNEY  OF  DINING-EOOM.  SECTION. 


507 


I       i       i 


J    FEET 


508 


ARCHITECTURAL  DRAWING. 


AROHITECTUEAL  DRAWING. 


509 


Figs.  1172  to  1177  are  plans  and  elevations  of  country  residences,  from 
Downing's  "Cottage  Houses." 

ELEVATION  OF  A  TIMBER  COTTAGE,  BY  GERVASE  WHEELER. 


FIG.  1172, 


510 


ARCHITECTURAL  DRAWING. 


The  construction  of  Fig.  1172,  though  simple,  is  somewhat  peculiar.  It  is 
framed  in  such  a  manner  that  the  construction  is  manifest  on  the  exterior. 
At  the  corners  are  heavy  posts,  roughly  dressed  and  chamfered,  and  into  them 
are  mortised  horizontal  ties,  immediately  under  the  springing  of  the  roof ; 


FIG.  1173. 


FIG.  1174. 


ENGLISH  EURAL  STYLE. 


FIG.  1175. 


ARCHITECTURAL  DRAWING 


511 


these,  with  the  posts  and  the  studs,  and  the  framing  of  the  roof,  show  exter- 
nally. Internally  are  nailed  horizontal  braces  at  equal  distances  apart,  stop- 
ping on  the  posts  and  studs  of  the  frame,  and  across  these  the  furring  and 
lathing  cross  diagonally  in  different  directions.  On  these  horizontal  braces, 
the  sheathing,  composed  of  plank  placed  in  a  perpendicular  position,  is  sup- 
ported and  retained  in  its  place  by  battens  two  and  a  half  inches  thick,  and 


RURAL  GOTHIC  STYLE.' 


FIG.  1176. 


512 


ARCHITECTURAL  DRAWING. 
ITALIAN  VILLA,  BY  UPJOHN. 


FIG.  1177. 


ARCHITECTURAL  D 


513 


made  with  a  broad  shoulder.      These  battens  are  pinned  to  the  horizonl 
braces,  confining  the  planks,  but  leaving  spaces  for  shrinking  and  swellfflfg, 
thus  preventing  the  necessity  of  a  single  nail  being  driven  through  the  planks. 
Fig.  1173  represents  the  batten,  B,  and  the  mode  of  framing. 

Fig.  1174  represents  the  usual  form  of  vertical  boarding,  which  is  less  ex- 
pensive than  the  first  illustration,  and,  in  general,  will  be  found  sufficiently 
secured  for  the  class  of  buildings  to  which  it  is  applied. 

Fig.  1178  represents  the  front  elevation  of  a  high-stoop  house  of  T.  Thomas 
design,  New  York  city. 

To  accommodate  the  poor  and  people  of  small  means  in  all  cities,  it  was, 
and  to  some  extent  still  is,  the  custom  to  divide  houses  which  were  intended 
for  single  occupation  into  small  apartments  for  many  families,  or  to  let  rooms 
singly  for  this  purpose.  This  was  found  to  be  objectionable  to  both  occupants 
and  owners,  and  houses  have  been  constructed  especially  for  the  poorer  classes. 
Virtually,  they  are  now  nearly  all  apartment-houses,  each  family  having  dis- 
tinct rooms  or  suites  to  itself.  But  the  term  tenement-houses  is  applied  to 
the  cheaper  kind  of  apartments,  occupied  by  the  poorer  class,  and  situated 
in  the  least  expensive  localities.  The  common  form  of  tenement-house  con- 
sists of  two  buildings,  one  in  the  front  and  one  in  the  rear  of  the  lot,  with  an 
outer  or  air  space  between.  A  hall  leads  through  the  first  story  to  the  central 
area ;  on  each  side  of  this  hall  there  may  be  small  stores  and  apartments. 
Stairs  from  the  hall  lead  to  the  apartments  above.  The  25  feet  is  divided  in 
two,  making  two  living-rooms  on  each  front ;  these  are  the  only  rooms  opening 
directly  into  the  outer  air.  Bedrooms  are  attached  to  each  of  these  rooms, 
but  take  their  light  and  air  from  the  staircases,  or  small  light-wells.  In  the 
rear  houses  there  are  two  tenements  to  each  story  ;  they  take  their  light  and 
air  from  the  central  and  back  areas.  Water-closets  or  privies  are  in  the  central 
area.  These  tenements  are  mostly  occupied  by  work-people,  largely  of  foreign 
birth,  dependent  directly  on  small  wages.  But  there  is  a  large  class,  of  limited 
means,  to  whom  these  accommodations  are  insufficient ;  parties  who  can  not 


well  afford  an  entire  house,  but  still  wish  for  the  privacy  of  one.     Within  the 
limits  of  a  lot  25'  X  100'  it  has  been  found  difficult  to  secure  all  the  necessaries 
of  light  and  ventilation,  with  the  number  of  suites  of  apartments  adapted  to 
the  means  of  the  occupants,  and  satisfactory  as  an  investment  to  the  owners. 
Fig.   1179  is  a  plan  of  one  of  the  best  of  these  designs.     It  provides  for 


83 


514 


ARCHITECTURAL  DRAWING. 


ARCHITECTURAL  DRAWING. 


515 


four  families  on  each  story,  although  it  will  be  observed  by  the  plan  of  the 
stairs  that  the  front  and  rear  tenements  are  not  on  the  same  flat ;  they  are 
separated  by  the  half  flight  of  stairs.  By  means  of  the  cross-shaped  court  be- 


PLAN. 


FIG.  1180. 


tween  the  adjacent  houses,  every  room,  including  the  bath-room,  has  a  window 
to  the  open  air.     This  is  the  most  commendable  feature  of  the  plan.     It  is 


516  ARCHITECTURAL  DRAWING. 

remarkable,  also,  however,  for  providing  more  conveniences  than  have  been 
customary  in  dwellings  of  this  class,  as,  for  instance,  a  small  bath-tub  as  well 
as  a  water-closet  for  each  family,  and  two  wash-tubs  as  well  as  a  sink  ;  also, 
a  dumb-waiter  (common  to  two  families)  for  bringing  up  fuel,  provisions,  etc. 
The  large  rooms  have  recesses  for  beds,  which  provide  for  an  extra  bedroom, 
while  detracting  but  little  from  their  value  as  parlors,  as  the  recess  may  be  cur- 
tained off  in  the  daytime,  or  the  bed  turned  up.  The  dimensions  of  the  rooms, 
as  marked  on  the  plans,  are  the  average  length  and  breadth.  These  suites  are 
much  too  restricted  for  a  very  large  class,  but  apartment-houses  somewhat  on 
this  model  are  constructed  in  desirable  localities,  where  the  accommodations 
and  conveniences  are  equal  to  those  of  any  private  house,  and  not  bounded  by 
the  limits  of  a  single  lot  nor  single  story,  many  unsurpassed  in  luxury  of  finish 
and  appointments. 

The  larger  apartment-houses  are  often  designated  as  French  flats,  or  flats. 
The  building  should  be  of  fire-resisting  construction.  The  suites  are  invariably 
supplied  with  water,  gas,  and  steam  heat ;  some  few  have  been  lighted  by  elec- 
tric light. 

Fig.  1180  is  an  illustration  of  a  "flat"  situated  on  the  corner  of  a  street, 
and  one  suite  takes  its  light  exteriorly  from  the  streets  while  the  other  depends 
in  a  measure  on  the  court.  Resistance  to  fire,  protection  from  vermin,  and 
privacy,  have  been  secured  by  the  absence  of  interior  light-wells  connecting 
stories,  solid  timbering  without  furring  or  framing  spaces.  Kitchens,  in  the 
figure,  are  attached  to  the  suites  ;  the  laundries  are  in  the  upper  story.  Many 
flats  are  without  kitchens  or  laundries,  and  meals  are  furnished  either  from 
without  or  from  restaurants  in  the  building.  It  then  corresponds  very  nearly 
to  a  hotel  without  transient  custom,  with  ample  and  separate  suites.  It 
would  seem  that  boarding-houses  might  be  built  on  such  plans — less  extensive 
in  their  arrangements  and  adapted  to  small  families  of  moderate  means ;  but 
boarding-houses  are  almost  invariably  private  houses,  but  little  modified  for  the 
more  public  use. 

Stores  and  Warehouses. — Fig.  1181  is  the  front  elevation  of  a  common 
type  of  New  York  city  store,  occupying  a  single  lot  of  25  ^!eet  in  width.  It 
will  be  observed  that  there  are  two  stories  beneath  the  level  of  the  sidewalk, 
the  basement  and  sub-cellar,  and  this  construction  still  obtains  largely  ;  but 
deep  basements  are  considered  preferable  by  some,  with  extra  stories  at  the 
top  rather  than  in  the  cellar.  Fig.  1182  is  a  section  of  the  front  wall,  showing 
heights  of  stories,  which  of  late  years  have  been  increased  over  former  practice, 
say  to  16'  for  the  first  story,  13'  for  the  second,  and  12'  and  11'  for  others,  the 
light  for  the  interior  being  taken  almost  universally  from  the  front  and  rear, 
and  skylights  done  away  with. 

Fig.  1183  is  a  plan  of  the  first-story  floor,  with  basement  in  front  dotted 
in  ;  five  feet  of  this  space,  or  that  usually  allotted  for  areas,  is  covered  with 
illuminating  tile  (Fig.  1184),  that  is,  small  glass  lenses,  set  in  iron  frames,  the 
whole  water-tight.  In  the  extreme  rear  there  is  a  small  area,  A,  open  to  the 
air,  of  about  5  feet,  for  light  and  air  to  the  basement  and  cellar.  The  offices  of 
the  first  story  are  situated  at  B,  over  which  there  is  usually  a  curved  lean-to  of 
illuminating  tile.  The  main  wall  above  this  story  is  on  the  line  a  I — plain 


ARCHITECTURAL   DRAWING. 


517 


SIDEWALK. 


FIG.  1181. 


FIG.  1182. 


518 


ARCHITECTURAL  DRAWING. 


brick — with  iron  shutters.     When  shutters  are  used  to  close  the  first-story 
front  they  are  mostly  rolling  shutters  of  sheet-steel.     The  hoist-way  to  the  up- 


FIG.  1183. 


per  stories  is  at  c,  a  position  somewhat  objectionable  as  interfering  with  the  use 
of  the  stairs,  when  a  common  hoist-wheel  is  used  ;  but  if  it  is  a  power-hoist, 
then  it  is  put  close  to  the  wall,  guarded  by  a  rail,  with  a  passage  round  to  the 


FIG.  1184. 


stairs.  In  50  feet  front  stores  the  hoist  is  put  on  the  opposite  corner  from  the 
stairs,  as  at  D,  but  this  cuts  off  considerable  light  from  the  first-story  front.  In 
some  the  arrangement  is  as  in  Fig.  1185,  in  which  the  hoists  c  c  are  in  the  rear  of 


.  L_ 


FIG.  1185. 


ARCHITECTURAL  DRAWING. 


519 


the  stairs.     The  arrangement  for  offices  in  the  rear  of  the  first  story  is  in  a  T, 
with  spaces  at  the  sides  for  the  ventilation  and  light  of  the  lower  stories.     It 


FIG.  1186. 


will  he  observed  that  there  is  no  central  door,  as  in  the  elevation  (Fig.  1181), 
which  last  most  usually  obtains  for  wholesale  stores.     For  retail  stores,  there 


520 


ARCHITECTURAL  DRAWING. 


are  usually  four  openings  in  the  25  feet,  as  shown  in  the  double  stores  (Fig. 
1186),  a  design  of  J.  B.  Snook. 


When  lots  are  only  100  feet  in  depth,  85  feet  can  be  utilized  by  the  building 
with  sufficient  light  from  the  ends,  but  very  often  the  stores  run  through  from 
street  to  street,  or  200  feet.  Formerly  the  central  portion  was  lighted  by  sky- 


ARCHITECTURAL  DRAWING.  521 

lights,  but  this  was  found  very  objectionable,  and  it  is  now  usual  to  leave  an 
open-air  shaft  on  one  side,  inclosed  by  brick  walls,  and  the  windows  protected 
by  iron  shutters.  The  space  should  be  30  to  40  feet  long  and  6  feet  wide, 
which  may  be  covered  in  the  first  story  with  glass.  If  this  recess  is  on  the 
side  occupied  by  the  staircases,  it  does  not  detract  from  the  inside  finish  of  the 
stores. 

Hoists  now  in  large  stores  are  power-hoists — that  is,  worked  by  either 
steam  or  water.  The  platform  of  a  freight-hoist  is  usually  5  feet  square  ;  for 
passenger-hoists,  in  wholesale  stores,  somewhat  less — 4'  X  5'.  For  the  raising 
of  goods  from  the  basement  or  sub-cellar  to  the  sidewalk  there  is  a  hatch  in 
the  front  light  platform,  opposite  some  window,  and  the  space  is  like  that 
of  freight-hoists,  5'  x  5'  ;  these  may  be  power  or  hand  hoists.  For  the  de- 
livery of  goods  into  these  stores  there  is  often  a  slide  or  incline,  iron-plated, 
ending  at  the  bottom  with  an  easy  curvo  to  the  horizontal,  down  which  boxes 
and  bales  are  slid. 

Fig.  1187  is  the  elevation  of  an  iron-front  store  100  feet  in  width,  among 
the  earliest  built  in  ^N'ew  York  city,  and  in  its  effect  is  as  satisfactory  as  any 
since  constructed. 

Fig.  1188  is  a  perspective  view  of  a  machine  and  blacksmith  shop,  built  by 
the  author  many  years  since .  It  was  built  for  a  purpose,  and  to  express  the 
purpose  constructionally  and  economically.  As  regards  convenience  and 
strength,  it  was  found  to  be,  on  occupation,  all  that  could  be  wished.  Some 
allowance  should  be  made  for  absence  of  color  in  the  sketch,  which  con- 
tributed much  to  architectural  effect.  Posts,  lintels,  window-frames,  sashes, 
and  ornamental  letters,  were  of  iron,  and  painted  a  very  deep  green  ;  the 
structure  was  of  brick,  with  sills  and  bands  of  rubbed  Ulster  bluestone,  roof 
of  Welsh  slate.  The  building  occupied  one  corner  of  Greene  and  Houston 
Streets,  in  this  city,  but  was  burned,  and  can  not,  therefore,  be  referred 
to  practically.  The  chimneys  shown  in  front,  although  not  dummies,  were 
never  used.  Power  and  heat  were  supplied  by  steam-boilers  in  the  front  vault, 
with  a  long,  slightly  inclined  flue  leading  to  a  chimney  at  the  center  of  the 
side  blank  wall.  On  each  side  of  this  chimney,  and  separated  by  a  thin 
with,  there  were  flues.  Forges  occupied  all  the  exterior  walls  of  the  base- 
ment, front  and  side  areas,  and  the  draught  was  upward  and  then  down  into 
the  nearly  horizontal  flues  connected  with  the  central  flues,  and  the  draught 
was  invariably  good.  Care  was  taken  that  all  angles,  horizontal  and  vertical, 
should  be  rounded. 

School- Houses. — Figs.  1189  and  1190  are  an  elevation  and  plan  of  a  country 
district  school-house,  with  seats  for  forty-eight  scholars.  There  are  two  en- 
trances, one  for  each  sex,  with  ample  accommodations  of  entry  or  lobby-room 
for  the  hanging  up  of  hats,  bonnets,  and  cloaks.  A  side  door  leads  from  each 
entry  into  distinct  yards,  and  an  inside  door  opens  into  the  school-room.  The 
desk,  T,  of  the  teacher,  is  central  between  the  doors,  on  a  platform,  P,  raised 
some  6"  or  8"  above  the  floor.  In  the  rear  of  the  teacher's  desk  is  a  closet  or 
small  room,  for  the  use  of  the  teacher.  The  seats  are  arranged  two  to  each 
desk,  with  two  alleys  of  18"  and  a  central  one  of  2'.  The  passages  around  the 
room  are  3'. 


522 


ARCHITECTURAL   DRAWING. 


ARCHITECTURAL  DRAWING. 


523 


H 

l 

• 

FIG.  1190. 

524 


ARCHITECTURAL  DRAWING. 


FIG.  1192. 


Figs.  1191  and  1192  are  the  elevation  in  perspective  and  plan  of  an  English 
country  school-house,  introduced  as  suggestive — whether  a  one-story  plan  might 
not  be  better  suited,  and  of  more  beautiful  effect  in  our  own  country  towns, 


CPLPCP 


where  there  is  plenty  of  ground  space,  than 
many  stories. 

On  the  Requirements  of  a  School- House. — Every  scholar  snoma  have  room 
enough  to  sit  at  ease,  his  seat  should  be  of  easy  access,  so  that  he  may  go  to 
and  fro,  or  be  approached  by  the  teacher  without  dis- 
turbing any  one  else.     The  seat  and  desk  should  be 
properly  proportioned  to  each  other  and  to  the  size  of 
the  scholar  for  whom  it  is  intended.     The  seats,  as         "1     I     I     I     I      |j 
furnished  by  the  different  makers  of  school  furniture, 
vary  from  9"  to  14"  in  height ;  and  the  benches  from 
17"  to  28" ;   measuring  on  the  side  next  the  scholar. 
The  average  width  of  the  desk  is  about  18",  and  it  is 
formed  with  a  slope  of  from  1£"  to  2-J",  with  a  small 
horizontal  piece  of  from  2"  to  3"  at  top.     There  is  a 
shelf  beneath  for  books,  but  it  should  not  come  within 
about  3"  of  the  front.     The  width  of  the  seat  varies 
from  10"  to  14",  with  a  sloping  back,  like  that  of  a  chair ; 

it  should,  in  fact,  be  a  comfortable  chair.    It  will  be  observed  that,  in  the  figure, 
two  scholars  occupy  one  bench.     Fig.  1193  represents  another  arrangement,  in 


FIG.  1193. 


SCHOOL  ROOM 
12.  x  BO. 


FIG.  1195. 


526 


ARCHITECTURAL  DRAWING. 


•it 


ARCHITECTURAL  DRAWING. 


527 


which  each  scholar  has  a  distinct  bench  ;  this  is  more  desirable,  but  not  quite  so 
economical  in  room.  In  primary  schools,  desks  are  not  necessary ;  and  in  many 
of  the  intermediate  schools  the  seat  of  one  bench  is  formed  against  the  back  of 
the  next  bench  ;  but  seats  distinct  are  preferable.  The  teacher's  seat  is  inva- 
riably on  a  raised  platform,  and  had  better  be  against  a  dead  wall  than  where 
there  are  windows.  Blackboards  and  maps  should  be  placed  along  the  walls. 
Care  should  be  taken  in  the  warming  and  ventilation  ;  warm  air  should  be  in- 
troduced in  proportion  to  the  number  of  scholars,  and  ventiducts  should  be 
formed  to  carry  off  the  impure  air. 

In  cities  and  large  towns  it  is  almost  indispensable  to  build  school-houses 
many  stories  in  height,  dividing  the  rooms  in  each  story  according  to  the  neces- 
sities of  their  occupancy.  The  management  of  schools  differs  in  different 
localities.  This  will  be  seen  in  the  illustrations  given  below,  showing  the  ar- 
rangements of  school-houses  in  the  city  of  New  York  and  of  Cleveland,  Ohio. 

Fig.  1194  is  an  elevation  in  perspective  of  one  of  the  largest  of  the  New  York 
city  schools,  showing  the  yards  around  it.  Fig.  1195  is  the  plan  of  the  gram- 


FIQ.  1196. 


mar-department  floors  of  this  house ;  and  Fig.  1196  the  plan  of  the  same  floors 
of  another  house  of  a  different  outline. 

Figs.  1197  to  1200  are  plans  of  school -houses,  built  at  Cleveland,  Ohio,  a 
type  inaugurated  under  the  supervision  of  the  then  superintendent,  Mr.  A.  J. 
Rickoff.  Figs.  1197,  1198,  and  1199  are  plans  of  the  High-School  house.  Fig. 


528 


ARCHITECTURAL   DRAWING. 


ARCHITECTURAL  DRAWING. 


529 


n  n 

n  D  DiJDJjn 

D  D  nib;  b  n  a 


a  D  DifDa  a 


D  D  Djiajja  D  a 
a  a  pijaiiq  a  a 


a  a  a  a  a  a  a 
a  a  D  a  a  a  a 
a  a  a  n  a 


a  a  n  a  a  a  a 
a  a  a  a  a  a  a 


n  D  D  D  n 


n-i  i  i  i  i  i  FEET 


34 


530  ARCHITECTURAL  DRAWING. 

1197  is  the  plan  of  the  third  story  ;  Figs.  1198  and  1199  of  those  portions  of 
the  second  and  first  stories  which  differ  from  that  of  the  third.  There  is  a 
rear  vestibule  in  the  first  story  to  correspond  with  the  one  in  front,  shown  in 
the  figure.  In  the  whole  building  there  are  14  session-rooms,  each  37'  X  30'  x 
16'  ;  each  having  its  connecting  cloak-room  ;  one  general  assembly-room,  94' X 
56'  X  38'  high,  with  a  seating  capacity  for  at  least  1,000  persons  ;  one  lecture- 
room,  with  seats  for  100,  with  an  apparatus-room  ;  one  room  for  drawing,  30' X 
55',  with  a  room  for  models,  drawing-boards,  etc.  ;  two  rooms  for  the  principal 
and  reception-room  ;  five  rooms  for  library  and  recitation-rooms. 

Fig.  1200,  a  plan  of  one  half  of  one  story  of  the  Walton  Avenue  School,  on 
a  larger  scale,  explains  more  fully  the  arrangement  of  seats  and  the  ventilation. 
Four  ventilating  educts,  of  8  square  feet  of  section  each,  may  be  heated  to  any 
required  temperature  for  the  purposes  of  circulation  by  four  upright  2"  steam- 
pipes  ;  six  ducts  of  1  square  foot  section  lead  from  different  points  in  the  floor 
of  each  session-room  (as  shown  in  dotted  lines  in  the  figure)  into  the  ventilating 
educts.  There  are  besides  other  registers  opening  directly  into  the  educts.  The 
building  is  heated  by  steam  coils  or  radiators  placed  under  the  windows  of  the 
rooms,  with  provision  for  the  admission  of  fresh  air  under  the  stone  sills  behind 
the  radiators.  It  will  be  observed  that  the  main  light  of  every  room  is  admitted 
at  the  left  hand  of  the  pupil,  so  that  in  writing  the  shadow  of  the  hand  does 
not  fall  on  the  space  to  be  written  on.  There  are  none  of  the  cross-lights 
that  so  seriously  impair  the  vision.  The  wall  facing  the  pupil  and  behind  the 
teacher  is  unbroken  by  windows,  aifording  large  and  convenient  spaces  for  black- 
boards. 

Churches,  Theatres,  Lecture- Rooms,  Music  and  Legislative  Halls. — To  the 
proper  construction  of  rooms  or  edifices  adapted  for  these  purposes  some  knowl- 
edge of  the  general  principles  of  acoustics,  and  their  practical  application,  is 
necessary.  In  the  case  of  lecture-rooms  and  churches,  the  positions  of  the 
speaker  and  the  audience  are  fixed  ;  in  theatres,  one  portion  of  the  inclosed 
space  is  devoted  to  numerous  speakers  and  the  other  to  the  audience  ;  in  legis- 
lative halls,  the  speakers  are  scattered  over  the  greater  part  of  the  space,  and 
also  form  the  audience. 

The  transmission  of  sound  is  by  vibrations,  illustrated  by  the  waves  formed 

by  a  stone  thrown  into  still  water  ;  but  direction  may  be  given  to  sound,  so  that 

the  transmission  is  not  equally  strong  in  every  direc- 

.,  - -  ^  tion;  thus,  Saunders  found  that  a  person  reading  at  the 

center  of  a  circle  of  100  feet  in  diameter,  in  an  open 
meadow,  was  heard  most  distinctly  in  front,  not  as  well 
at  the  sides,  but  scarcely  at  all  behind.  Fig.  1201 
shows  the  extreme  distance  every  way  at  which  the  voice 
could  be  distinctly  heard  :  92  feet  in  front,  75  feet  on 
each  side,  and  31  feet  in  the  rear.  The  waves  of  sound 
are  subject  to  the  same  laws  as  those  of  light,  the  angles 
FIG.  12Q1.  Of  reflection  are  equal  to  those  of  incidence  ;  therefore, 

in  every  inclosed  space  there  are  reflected  sounds,  more 

or  less  distinct,  according  to  the  position  of  the  hearer,  and  to  the  form  and 
condition  of  the  surfaces  against  which  the  waves  of  sound  impinge.  Thus, 


ARCHITECTURAL  DRAWING. 


531 


•of  all  the  sounds  entering  a  parabolic  sphere,  the  reflected  sounds  are  collected 
at  the  focus.  Solid  bodies  reflect  sound,  but  draperies  absorb  it.  As,  in  all 
rooms,  the  audience  can  never  be  concentrated  at  focal  points,  nor  is  it  pos- 
sible in  any  construction  to  make  calculation  for  all  positions,  it  is  in  general 
best  to  depend  on  nothing  but  the  direct  force  of  the  voice,  and  not  to  con- 
struct larger  than  can  be  heard  directly  without  aids  from  reflected  sounds. 

There  is  great  difference  in  the  strength  of  voice  of  different  speakers  ;  the 
limits  as  given  in  the  figure  are  for  ordinary  reading  in  an  open  space.  In  in- 
closed spaces,  owing  to  the  reflected  sounds  or  some  other  cause,  there  are  cer- 
tain pitches  or  keys  peculiar  to  every  room,  and  to  speak  with  ease  the  speaker 
must  adapt  his  tone  to  those  keys.  The  larger  the  room,  the  slower  and  more 
distinct  should  be  the  articulation. 

It  has  been  observed  that  the  direction  of  the  sound  influences  the  extent  to 
which  it  may  be  heard.  The  direction  of  the  currents  of  air  through  which 
the  sound  passes  affects  the  transmission  of  the  sound,  and  this  may  be  made 
useful  when  the  rooms  are  heated  by  hot  air,  by  introducing  the  air  near  the 
speaker,  and  placing  the  ventilators  or  educts  at  the  outside  of  the  rooms,  and 
by  placing  their  apertures  rather  nearer  the  bottom  of  the  room  than  at  the 
top.  It  would  seem  much  better  and  easier  to  make  a  current  of  air  a  vehicle 
of  sound  rather  than  depend  on  reflection. 

On  the  Space  occupied  by  Seats  in  general. — A  convenient  arm-chair  occu- 
pies about  20"  X  20*,  the  seat  itself  being  about  18"  in  depth,  and  the  slope  of  the 
back  2" ;  18"  more  affords  ample 
space  for  passage  in  front  of  the 
sitter.  In  churches  the  seats  are 
arranged  by  pews  or  stalls  ;  the 
width  of  each  pew  in  general  being 
about  2'  10".  In  the  arrangement 
of  seats  at  the  Academy  of  Music 
the  bottom  turns  up  (Figs.  1202 
and  1203),  and  29"  only  is  allowed 
for  both  seat  and  passage-way,  and 
18"  for  the  width  of  seat,  which 
may  be  taken  as  the  average  allow- 
ance in  width  to  each  sitter  in 
comfortable  public  rooms.  In  lec- 
ture-rooms, benches  and  settees  are  often  used,  the  space  there  occupied  by 
seat  and  passage  being  about  2'  6". 

In  the  earlier  churches,  ceremonies  and  rites  formed  a  very  large  part  of  the 
worship,  the  sight  was  rather  appealed  to  than  the  hearing,  and  for  this  pur- 
pose churches  were  constructed  of  immense  size,  and  with  all  the  appliances  of 
ornament  and  construction,  with  pillars,  vaults,  groins,  and  traceried  windows. 
In  the  churches  of  this  country,  the  great  controlling  principle  in  the  construc- 
tion of  a  church  is  its  adaptation  to  the  comfortable  hearing  and  seeing  the 
preacher.  In  this  view  alone,  the  church  is  but  a  lecture-room  ;  but  since  even 
the  character  of  the  building  may  tend  to  devotional  feelings  in  the  audience, 
.and  since  certain  styles  and  forms  of  architecture  have  long  been  used  for  church 


FIG.  1202. 


FIG.  1203. 


532 


ARCHITECTURAL  DRAWING. 


edifices,  and  seem  particularly  adapted  for  this  purpose,  it  has  been  the  custom 
to  follow  these  time-honored  examples,  adapting  them  to  the  modern  require- 
ments of  church  worship. 

Fig.  1205  is  a  plan  of  an  ancient  basilicon  or  Romanesque  church.  Fig. 
1204  is  a  sectional  elevation  of  the  same.  Fig.  1206  is  a  plan  of  a  Gothic 
church,  in  which  C  is  the  chancel,  usually  at  the  eastern  extremity,  T  T  the 
transept,  and  N  the  nave.  In  general  elevation  the  Gothic  and  Romanesque 
agree  :  a  high  central  nave  and  low  side  aisles.  In  the  later  Romanesque  the 
transept  is  also  added. 


FIG.  1204. 


FIG.  1205. 


The  basilicas  aggregated  within  themselves  all  the  offices  of  the  Romish 
church.  The  circular  end  or  apse,  and  the  raised  platform,  or  dais,  in  front  of 
it,  was  appropriated  entirely  to  the  clergy ;  beneath  was  the  crypt  or  confes- 
sional, where  were  placed  the  bodies  of  the  saints  and  martyrs,  and  pulpits  were 
placed  in  the  nave,  from  which  the  services  were  said  or  sung  by  the  inferior 
order  of  clergy. 

The  plan  (Fig.  1206)  is  that  of  the  original  Latin  cross,  the  eastern  limb 
or  chancel  being  the  shortest,  and  the  nave  the  longest.  Sometimes  the  eastern 
limb  was  made  equal  to  that  of  the  transept,  sometimes  even  longer,  but  never 
to  exceed  that  of  the  nave.  In  the  Greek  cross  all  the  limbs  are  equal.  In 
most  of  the  French  Gothic  churches  the  eastern  end  is  made  semicircular,  often 
inclosed  by  three  or  more  apsidal  chapels,  that  is,  semi-cylinders,  surmounted 
by  semi-domes. 

The  Byzantine  church  consisted  internally  of  a  large  square  or  rectan- 
gular chamber,  surmounted  in  the  center  by  a  dome,  which  rested  upon 
massive  piers ;  an  apse  was  formed  at  the  eastern  end.  Circular  churches 
were  built  in  the  earlier  ages  for  baptisteries,  and  for  the  tombs  of  saints  and 
emperors. 

The  Greek,  Roman,  and  English  churches  conform  in  their  cathedrals  and 
larger  edifices  nearly  to  the  Romanesque  or  Gothic  models.  But  as  the  general 
requirements  for  church  services  now  are  those  of  a  lecture-room — comfortable 
seats,  convenient  for  hearing  and  seeing  the  preacher,  with  adequate  means  of 
heating  and  ventilation,  for  which  the  older  forms  are  not  suited — modern 
churches  are  constructed  adapted  to  these  purposes,  and,  in  cities,  to  the  size 
and  form  of  the  lots,  with  some  ecclesiastical  accessories  of  towers  and  steeples: 
windows  and  doors  and  interior  finish. 


AECHITEOTURAL  DRAWING. 


533 


534 


ARCHITECTURAL  DRAWING. 


Figs.  1207  and  1208  are  the  elevation  and  plan  of  a  London  Wesleyan 
chapel  characteristic  of  the  above. 


FIG.  1209. 


Figs.  1209  and  1210  are  the  elevations  and  plan  of  the  English  church  at 
the  Hague,  where  aesthetic  effect  has  been  more  studied  than  in  the  above  ex- 
ample, with  less  economy  in  the  occupancy  of  the  lot. 


ARCHITECTURAL  DEAWI 

The  length  of  pews  is  various,  being  generally  of 
small  or  large  families,  say  from  7'  6"  to  11'  6",  IS" 
ter.  In  arrangement  it  is  always  considered  desirable 


#sjzes,  adapted  to  either 
allowed  for  each  sit- 


there  should  be  a 


FIG.  1210. 

central  aisle,  and  if  but  four  rows  of  pews,  two  aisles  against  the  wall ;  if  six 
rows,  one  row  on  each  side  will  be  wall-pews.  Formerly  it  was  the  universal 
practice  to  construct  pews  with  doors,  but  of  late  it  is  more  customary  to  omit 
the  doors,  making  the  pews  open  stalls. 

Few  churches  are  now  without  an  organ  ;  its  dimensions  should  of  course 
depend  on  the  size  of  the  church.  In  form  it  may  be  adapted  somewhat  to 
the  place  which  may  be  appropriated  to  it — either  in  a  gallery  over  the  main 
entrance,  or  at  the  side  of  the  chancel,  as  in  Fig.  1210.  In  general,  it  is  ob- 
long in  form,  the  longer  side  being  with  the  keys.  The  dimensions  suited  to 
a  medium-sized  church  are  about  9'  X  15',  and  12'  in  height. 

The  vestry-room,  if  used  for  the  purposes  of  its  meetings,  should  be  adapted 
in  size  to  the  purpose  ;  but  if  only  for  a  withdrawing  or  robing  room  for  the 
clergyman,  it  may  be  of  very  small  dimensions,  and  should  be  accessible  from 
without.  The  Sunday-school  room,  in  general,  requires  in  plan  about  half 
the  area  of  the  church.  From  motives  of  economy  it  is  usually  placed  in  the 
basement  of  the  church  ;  bufc,  in  the  country  especially,  it  is  better  that  it 
should  be  a  separate  building,  and  form  one  of  the  group  of  church,  parson- 
age, and  Sunday-school  house. 

In  elevation,  city  churches  are  Greek  with  porticoes  in  front,  Romanesque, 
and  Gothic,  occasionally  Byzantine.  The  Greek  have  no  tower,  but  often  a 
spire  above  the  portico  ;  the  Romanesque  and  Gothic  generally  one  tower,  over 
the  central  door  of  entrance,  or  at  one  corner  ;  sometimes  two,  one  at  each  side 
of  the  principal  door,  almost  invariably  surmounted  by  spires,  high  and  taper- 
ing, usually  of  wood,  but  in  some  instances  of  stone. 

Fig.  1211  is  the  front  elevation  of  the  Roman  Catholic  cathedral  in  Fifth 
avenue,  New  York  city,  from  designs  by  James  Renwick,  architect.  The  style 
is  the  French  Decorated  Gothic. 

Fig.  1212  is  a  perspective  view  of  the  Episcopal  church  of  St.  Bartholomew, 
corner  of  Forty-fourth  Street  and  Madison  Avenue,  New  York  ;  Renwick  and 
Sands,  architects.  The  style  is  Romanesque  ;  the  vestry  and  parsonage  are  con- 
nected with  the  church. 


536 


ARCHITECTURAL  DRAWING. 


FIG.  1211. 


ARCHITECTURAL  DRAWING. 


537 


FIG.  1212. 


538 


ARCHITECTURAL  DRAWING. 


Fig.  1213  is  the  cross-section  of  a  common  form  of  small  country  church, 
with  nave  n,  aisles  a  a,  and  clear-story  c.     The  effect,  both  inside  and  out,  is 


FIG.  1213. 

good,  but  there  are  objections  to  the  masonry-columns,  which  cut  off  the  view 
of  the  desk  and  the  altar  from  many  sitters,  and  to  the  windows  of  the  clear- 
story, that  in  the  winter  they  act 
as  coolers  to  the  air  which  de- 
scends in  draughts  upon  the  heads 
of  the  congregation  beneath  them. 
Neither  columns  nor  clear-story 
are  constructively  necessary ;  the 
span  can  readily  be  met  by  a  sin- 
gle roof,  and  sufficient  light  can 
be  obtained  from  the  sides. 

Figs.  1214,  1215,  and  1216 
are  examples  of  open-timbered 
Gothic  roofs  of  churches. 

The    technical    names    (Fig. 
1214.  1214)  are  :  1,  Principals  ;  2,  Pur- 


AKCHITECTURAL  DRAWING. 


530 


1215. 


lines  ;   3,  Collars ;   4,  Braces  ;   5,  Wall-pieces  ;    6,  Wall-plates  ;    7,  Struts  ;   8, 
Rafters.     4  and  5  are  shown  in  section. 

Theatres. — In  theatres  and 
opera-houses  it  is  not  only  ne- 
cessary that  the  audience  should 
have  a  good  position  for  hear- 
ing and  seeing  the  performance 
upon  the  stage,  but  also  to  see 
each  other.  The  most  approved 
form,  now,  for  the  body  of  a 
dramatic  theatre  is  a  circular 
plan,  the  opening  -for  the  stage 
occupying  from  one  fourth  to 
one  fifth  of  the  circumference, 
the  sides  of  the  proscenium  be- 
ing short  tangents  ;  but  for  a 
lyric  theatre,  where  music  only 
is  performed,  and  where,  conse- 
quently, hearing  is  easier,  the 
curve  is  elongated  into  an  ellipse, 
with  its  major  axis  toward  the 
stage. 

In  the  general  position  of 
the  stage,  proscenium,  orches- 
tra, orchestra-seats,  parquette, 
and  boxes,  but  one  plan  is  fol- 
lowed. The  line  of  the  front 
of  the  stage,  at  the  foot-lights, 
is  generally  slightly  curved,  with 
a  sweep,  say,  equal  to  the  depth 
of  the  stage,  and  the  orchestra 
and  parquette  seats  are  arranged 
in  circles  concentric  with  it :  of 
the  space  occupied  by  seats  we 

have  already  spoken.  The  entrance  to  the  parquette  may  be  through  the  boxes, 
near  the  proscenium,  and  centrally,  but  better  at  the  sides,  dividing  the  boxes 
into  three  equal  benches  ;  the  seats  in  the  boxes  are  usually  concentric  with  the 
walls,  and  more  roomy  than  those  of  the  parquette.  The  orchestra  seats  are  of 
a  height  to  bring  the  shoulders  of  the  sitter  level  with  the  floor  of  the  stage, 
and  the  floor  of  the  parquette  rises  to  the  outside,  1  in  15  to  18.  The  floor  of 
the  first  row  of  boxes  is  some  2  to  3  feet  above  the  floor  of  the  parquette  at  the 
front  center,  and  rises,  by  steps  at  each  row,  some  4  inches  ;  in  the  next  tier  of 
boxes  the  steps  are  considerably  more  in  height,  and  so  on  in  the  boxes  above. 
In  general,  three  rows  of  boxes  are  all  that  is  necessary  ;  in  front,  above  the 
second,  the  view  of  the  stage  is  almost  a  bird's-eye  view.  The  floor  of  the 
Btage  descends  to  the  foot-lights  at  the  rate  of  about  1  in  50.  In  large  theatres 
it  is  of  the  utmost  importance  that  all  the  lobbies  or  entries  should  be  spacious, 


JTia.  1216. 


540 


ARCHITECTURAL  DRAWING. 


FIG.  1217. 


and  the  means  of  exit  numerous  and   ample — the   staircases  broad,  in  short 

flights  and  square  landings,  and  not  circular,  as,  in  case  of  fright,  the  pressure 

of  persons  behind  may  precipitate  those 
in  front  the  whole  length  of  the  flight. 
Ladies'  drawing-rooms  should  be  placed 
convenient  to  the  lobbies,  of  a  size 
adapted  to  that  of  the  theatre,  arranged 
with  water-closets  ;  there  should  also 
be  provided  rooms  for  the  reception  of 
gentlemen's  canes  and  umbrellas,  with 
water-closets  attached.  The  box-office 
should  be,  of  course,  near  the  entrance, 
but  so  arranged  as  to  interfere  as  little 
as  possible  with  the  approach  to  the 
doors  of  the  house.  At  the  entrance 
there  should  be  a  very  spacious  lobby, 
or  hall,  so  that  the  audience  may  wait 
sheltered  from  the  weather ;  if  possi- 
ble, there  should  be  a  long  portico  over 
the  sidewalk,  to  cover  the  approach  to 

the  carriages.     Only  single  entrances  are  necessary  to  distinct  parts  of  the 

house,  but  the  greater  the 

number  of,  and  the  more  PLAN. 

ample  places  for  exit  at 

the  conclusion  of  the  piece, 

or  for  the  contingency  of 

fire,  the  better. 

Fig.    1217   is   a   plan 

suggested  by  Ferguson  of 

keeping  the  center  of  the 

boxes  perpendicular  over 

one  another,  and  then,  by 

throwing  back  each  tier 

of  side-boxes  till  the  last 

is  a  semicircle,  the  whole 

audience  would   sit  more 

directly  facing  the  stage, 

would  look  at  it  at  a  bet- 
ter angle,  and  the  volume 

of  sound  be  considerably 

increased  by  its  freer  ex- 
pansion   immediately   on 

leaving  the  stage. 

Fig.  1218  and  1219  are 

a  plan  and  section  of  Wag- 
ner's theatre. 

In  cities,  the  auditoria 


FIG.  1219. 


AKCHITECTUKAL  DRAWING. 


541 


of  dramatic  theatres  conforming  to  the  shape  of  the  lots  are  rectangular  in 
their  outline,  and  seldom  exceed  a  seating  capacity  of  1,000.  Lyric  theatres 
are  much  larger,  seating  often  as  many  as  2,000,  and  conforming  in  their 
interior  outline  to  the  art  requirements.  Lecture-rooms  are  usually  arranged 
with  the  audience-floor  flat,  room  rectangular,  with  reading-desk  or  platform 
raised,  and  with  or  without  galleries.  The  same  form  usually  obtains  for 
music-halls,  only  they  are  much  greater  in  extent ;  the  first  being  capable  of 
containing  from  500  to  800  persons ;  whereas  some  music-halls  will  contain 
2,000,  and  Ferguson  thinks  that  a  music-hall  might  be  arranged  so  that  even 
10,000  might  hear  as  well  as  in  those  of  present  construction.  The  lecture 
and  music  halls  are  seldom  devoted  to  a  single  purpose,  but  are  used  for 
political  meetings,  for  fairs,  and  dances,  and  the  construction  must  be  such  as 
to  serve  these  other  purposes. 

COMPARATIVE  TABLE  OF  THE  DIMENSIONS   OF  A  FEW  THEATRES. 


DISTANCE 

IN   FEET 

HEIGHT, 

IN  FEET. 

NAME  AND  LOCATION. 

Between  boxes 
and  footlights. 

Between  footlights 
and  curtain. 

Between  curtain 
and  back  of  stage. 

Greatest  breadth 
of  pit. 

Breadth  of  cur- 
tain. 

Breadth  of  stage 
between  side-walls. 

it. 

|| 

If 
o| 

§0 

11 

Alexandra,  St.  Petersburg  

65 

11 

84 

58 

56 

75 

53 

58 

,  Berlin  

62 

16 

76 

51 

41 

92 

43 

47 

La  Scala,  Milan  .... 

77 

18 

78 

71 

49 

86 

60 

64 

San  Carlo,  Naples  

77 

18 

74 

74 

52 

66 

81 

83 

Grand  Theatre,  Bordeaux  

46 

10 

69 

47 

37 

80 

50 

57 

Salle  Lepelletier,  Paris  

67 

9 

82 

66 

43 

78 

52 

66 

Covent  Garden,  London 

66* 

55 

51 

32 

86 

54 

Drury  Lane,  London. 

64* 

80 

56 

32 

48 

60 

Boston,  Boston  

53 

18 

68 

46 

87 

554 

58 

Academy  of  Music,  New  York  

74 

13 

71 

62 

48 

83 

74 

Grand  Opera-  House,  New  York  

54 

84 

63i 

48 

44 

?6 

52 

67 

Opera-House,  Philadelphia 

61 

17 

72 

66 

48 

90 

644 
~ 

74 

*  These  dimensions  include  the  distance  between  the  footlights  and  curtain. 

Legislative  Halls. — Although  much  has  been  written  about  their  construc- 
tion in  relation  to  acoustic  principles,  there  yet  seems  to  be  great  disagreement 
in  practical  examples,  and  in  the  deductions  of  scientific  men.  The  Chamber 
of  French  Deputies  was  constructed  after  a  report  of  most  celebrated  architects, 
in  a  semicircular  form,  surmounted  by  a  flat  dome,  but  as  the  member  inva- 
riably addresses  the  house  from  the  tribune,  at  the  center,  in  its  requirements 
it  is  but  a  lecture-room.  Mr.  Mills,  architect,  of  Philadelphia,  recommends 
for  legislative  or  forensic  debate,  a  room  circular  in  its  plan,  with  a  very  slightly 
concave  ceiling.  Dr.  Eeid,  on  the  contrary,  in  reference  to  the  Houses  of  Par- 
liament, gave  preference  to  the  square  form,  with  a  low,  arched  ceiling.  The 
Hall  of  Representatives,  at  Washington,  is  139  feet  long  by  93  feet  wide,  and 
about  36  feet  high,  with  a  spacious  retiring  gallery  on  three  sides,  and  a  re- 
porters' gallery  behind  the  Speaker's  chair.  The  members'  desks  are  arranged 


542  ARCHITECTURAL   DRAWIXG. 

in  a  semicircular  form.  The  ceiling  is  flat,  with  deep-sunk  panels,  openings 
for  ventilation,  and  glazed  apertures  for  the  admission  of  light.  The  ventila- 
tion is  intended,  in  a  measure,  to  assist  the  phonetic  capacity  of  the  hall,  the 
air  being  forced  in  at  the  ceiling  and  drawn  out  at  the  bottom. 

In  reviewing  the  general  principles  of  acoustics,  it  will  be  found  that  those 
rooms  are  the  best  for  hearing  in  which  the  sound  arrives  directly  to  the  ear, 
without  reflection  ;  that  the  sides  of  the  room  should  neither  be  reflectors  nor 
sounding-boards,  and  that  surfaces  absorbing  sound  are  less  injurious  than  those 
that  reflect.  Slight  projections,  such  as  ornaments  of  the  cornices  and  shallow 
pilasters,  tend  to  destroy  sound,  but  deep  alcoves  and  recessed  rooms  produce 
echoes.  Let  the  ceiling  be  as  low  as  possible,  and  slightly  arched  or  domed  ; 
all  large  external  openings  should  be  closed  ;  as  M.  Meynedier  expresses  it,  in 
his  description  of  an  opera-house,  "Let  the  hall  devour  the  sound;  as  it  is 
born  there,  let  it  die  there." 

Hospitals. — In  large  cities,  hospitals,  by  necessity,  are  confined  to  narrow 
spaces,  but  they  should  be  placed,  if  possible,  on  river  fronts  or  on  open  parks, 
to  secure  as  much  open-air  ventilation  as  possible.  They  are  usually  many  sto- 
ries in  height,  with  large  wards  one  above  the  other.  Sir  J.  T.  Simpson  alleges 
a  very  high  rate  of  mortality  in  hospitals  after  surgical  operations  as  compared 
with  the  mortality  after  the  same  operations  wheu  performed  at  the  homes  of 
the  patients,  and  asserts  that  the  mortality  after  operations  performed  in  hos- 
pitals containing  more  than  300  beds  is  in  excess  of  that  in  hospitals  containing 
less  ;  that  great  hospitals  are  great  evils  in  exact  proportion  to  their  magnitude, 
and  suggests  the  construction  of  smaller  hospitals. 

Figs.  1220  and  1221  are  an  elevation  and  plan  of  an  English  country  hos- 
pital. 

Stables. — Under  this  general  name  are  included  the  barn,  or  the  receptacle 
of  hay  and  fodder,  the  carriage-house,  and  the  stable  proper,  or  lodging-house 
for  horses  and  cows.  The  first  two  may  be  included  under  one  roof,  the  car- 
riages on  the  first  floor,  and  hay  in  the  loft ;  but  the  lodging-place  should  be 
distinct,  in  a  wing  attached  to  the  barn,  that  the  odors  from  the  animals  may 
not  impregnate  their  food,  or  the  cloth-work' of  the  carriages,  or  the  ammonia 
tarnish  their  mountings. 

Hay  in  bulk,  in  the  mow,  occupies  about  340  cubic  feet  per  ton  ;  bales  aver- 
age 2'  4"  x  2'  6"  x  4',  and  weigh  from  220  to  320  pounds.  The  door-space  for 
a  load  of  hay  in  the  bulk  should  be  from  12  to  13  feet  high  and  12  feet  wide. 
The  floor  beneath  the  hay  should  be  tight,  so  that  dust  and  seed  may  not  drop 
on  the  carriage.  A  door  for  carriages  should  be  10  feet  6  inches  high  by  9  feet 
wide. 

The  horse  is  to  be  treated  with  greater  care  than  any  other  domestic  ani- 
mal. His  stable  is  to  be  carefully  ventilated,  that  he  may  have  fresh  air  without 
being  subject  to  cross-draughts.  Preferably,  the  floor  should  be  on  the  ground, 
that  there  may  be  no  cold  from  beneath .  He  should  stand  as  near  as  possible 
level  ;  and  for  this  purpose  a  grated  removable  floor,  with  small  interstices, 
should  be  laid  over  a  concrete  bottom,  with  a  drip  toward  the  rear  of  the  stall, 
and  the  urine  should  be  collected  in  a  drain,  and  discharged  into  a  trapped 
manure-tank  outside  the  stable.  In  Fig.  1222  the  pitch  of  bottom  of  stalls  is 


ARCHITECTURAL   DRAWING. 


543 


FIG.  1220. 
GROUND   PLAN. 


FIG.  1221. 


544 


ARCHITECTURAL  DRAWING. 


to  the  center  and  outward.  The  manure  should  never  be  deposited  beneath 
the  stable,  but  should  be  wheeled  out  and  deposited  in  a  manure-yard  or  tank 
daily.  It  is  as  essential  that  all  excrements  should  be  removed  entirely  from 
the  stable  as  that  the  privy  should  be  placed  outside  the  house. 

The  breadth  of  stalls  should  be  from  4  feet  6  inches  to  5  feet  in  the  clear ; 
the  length,  7  feet  6  inches  to  8  feet ;  the  rack  and  feed-box  require  two  feet  in 
addition,  to  which  access  is  given  in  the  best  stables  by  a  passage  in  front. 
Rack  and  feed-boxes  are  often  made  of  iron,  and  the  upper  part  of  stalls  fitted 
with  wrought-iron  guards.  Box-stalls,  in  which  horses  are  shut  up  but  not 
tied  in  cases  of  sickness  or  foaling,  are  about  10  feet  square, 


FIG.  1222. 


In  large  stables  in  cities  the  first  floors  are  often  occupied  by  the  carriages, 
while  the  horse-stalls  are  in  the  basement  or  upper  stories,  with  inclined  ways 
of  access.  In  the  basement  provision  must  be  made  for  light  and  ventilation. 


Tool 
House- 


Open      Shade 


0 

o 

0 

0 

Box    St&Us. 

Carriage    House 


FIG.  1223. 


ARCHITECTURAL  DRAWING. 


545 


In  the  upper  stories  these  may  be  secured  more  readily,  but  the  floors  must  be 
made  tight  and  deafened,  that  the  urine  may  not  leak  through,  nor  the  cold 
come  through  from  below  to  make  too  cool  a  bed  for  the  horse. 

Fig.  1222  is  an  elevation  in  perspective  of  two  first-class  stalls,  a  box  shown 
with  the  door  open,  and  a  single  stall.     The  lower  part  of  the  inclosures  is  of 
plank,  with  wrought-iron  guards  and  ramp  above.    The  posts  are  of  oak, 
and  the  hay-boxes  or  mangers  of  cast-iron  ;  the  hay-rack  in  the  box-stall 
is  of  wrought-iron.     These  are  of  common  manufacture,  and  are  of 
varied  patterns  ;  but  in  the  country  they  are  usually  made  of  wood, 
and  connected  with  the  stall. 

Fig.  1223  is  the  plan  of  a  small  country  stable,  show- 
ing the  desirable  passages  around  the  stalls  and  ex- 
terior windows  in  front  of  each  stall,  that 
the  horses  may  not  only  have  light 
and  air,  but  can  see  out. 

Coiv  -  houses,    for    cows 
giving  milk,  should 
be  constructed 
with  care 


FIG.  1224. 


FEET 


for  ventilation,  light,  and  cleanliness.  Other  cattle  are  usually  left  out,  with 
sheds  under  which  they  can  go  for  shelter.  For  those  housed,  the  spaces  occu- 
pied should  be  about  the  same  per  Head  as  the  single  horse-stall.  The  manger 


35 


546 


ARCHITECTURAL  DRAWING. 


should  be  on  the  floor,  12"  to  18"  high,  and  about  18"  wide.  It  is  not 
usual  to  have  partitions,  but  there  ought  to  be  between  every  pair, 
reaching  from  the  manger  half-way  to  the  gutter  behind.  The 
floor  should  be  level,  grated,  with  a  drip  beneath,  and  cleansed 
by  washing  out.  In  England  the  partition  and  man- 
gers are  often  of  cast-iron,  and  are  on  sale,  but 
here  they  are  of  wood. 

Greenhouses. — Fig.  1224:  is 
section  of  a  greenhouse,  with 
shelves  for  plants.    The 


the 


{FEET. 


FIG.  1225. 


ARCHITECTURAL  DRA\V|N£.  547 


floor  is  of  concrete  and  the  walls  are  of  masonry  ;  E^oiorthern  exposure  is  a 
blank  wall. 

Fig.  1225  are  the  details  of  windows.  The  sides  are  box-sash,  hung  with 
weights  (w,  w,  Fig.  1226).  The  lower  roof  sash  is  firmly  fixed,  but  the 
upper  one  can  be  slid  down  ;  it  is  usually  retained  in  place  by  a  cord  attached 
to  the  lower  part  of  the  sash,  passing  over  a  pulley  on  the  upper  bar  of  the 
frame,  with  the  loose  end  within  reach  of  the  gardener,  who  can  fasten  it  to 
a  cleat. 

Ventilation  and  Warming.  —  The  purposes  of  ventilation  are  not  changes  of 
air  merely,  but  the  removal  of  foul  and  vitiated  air,  and  the  substitution  there- 
for of  pure  air  ;  and  this  air  may  be  warm  or  cool  according  to  the  necessities 
of  the  season  and  personal  requirements.  Open  space  is  not  necessarily  well 
ventilated  ;  there  must  be  circulation,  outward  and  inward,  the  latter  from 
purer  sources  than  the  former,  or  the  change  is  useless.  With  an  equal  dis- 
charge and  supply  of  pure  air,  the  smaller  the  room,  the  more  frequent  the 
change  of  air,  the  better  its  distribution,  and  the  better  the  ventilation.  But 
if  the  means  of  removal,  supply,  and  distribution  of  air  be  proportioned  to  the 
size  of  the  room,  then  the  larger  the  room  the  better.  Apertures  do  not  neces- 
sarily mean  circulation  ;  a  flue  may  draw  or  it  may  not  draw,  it  may  be  inert, 
or  the  air  may  come  down  ;  a  window  may  be  open,  with  little  or  no  inward  or 
outward  movement  of  air.  In  a  house  exposed  to  a  fresh  breeze,  on  the  wind- 
ward side  there  is  an  air-pressure  ;  on  the  leeward  side  there  is  an  eddy  or 
vacuum.  Air  is  forced  in  on  the  first  through  every  crack  of  door  and  win- 
dow —  often  down  chimney-flues  —  and  drawn  out  on  the  other  side.  This  often 
happens  even  with  fires  in  the  chimneys,  and  with  heat  in  ventilating  educts. 
If  one  will  make  an  experiment  in  cold  weather,  when  the  windows  are  closed, 
and  there  are  fires  in  some  rooms,  he  will  find  that  there  is  cold  air  coming 
down  the  unused  flues,  and  will  feel  the  cold  current  flowing  down  the  stairs, 
and  along  the  floors  to  the  fires.  Architects  have  placed  kitchens  in  the  base- 
ment, and  in  the  attic,  and  the  smell  of  cooking  will  rise  through  the  house, 
usually  from  the  one,  but  descend  from  the  other  when  the  air  is  light  and 
muggy. 

Every  room  should  have  its  separate  flue  ;  for  if  the  current  is  not  upward 
it  will  probably  be  downward,  affording  a  fresh  supply  if  there  is  an  exit  else- 
where. A  chimney-flue  may  be  too  large  for  the  purposes  of  a  fire  ;  for  most 
fires  a  flue  8"  X  8"  is  amply  sufficient,  and,  for  the  purposes  of  ventilation  in 
the  common  occupation  of  a  house,  this  flue  will  answer  all  the  purposes  in 
cold  weather.  It  is  usual  to  depend  largely  on  windows  for  ventilation,  but 
the  space  on  which  they  open  may  be  too  circumscribed  to  afford  the  requisite 
change  of  air,  or  the  outer  air  itself  may  be  too  hot,  or  too  cold,  or  too  mala- 
rial or  offensive,  to  make  the  change  of  air  sanitary  or  pleasant.  In  tenement 
or  apartment  houses  care  should  especially  be  taken  that  the  inner  windows 
on  different  flats  open  into  as  large  air-shafts  as  possible,  and  that  these 
shafts  should  l^ave  free  opening  to  the  outer  air  without  sky-lights  ;  and  that 
the  floors  should  be  tight,  so  that  the  smells  may  not  pass  from  one  flat  to 
another.  Nothing  more  surely  shows  faults  in  ventilation  than  the  diffusion 
of  kitchen-smells  or  tobacco-smoke.  For  the  separation  of  apartments,  let 


548  ARCHITECTURAL  DRAWING. 

every  room  have  its  own  flue,  and  this  flue  extending  independently  well  above 
the  roof,  and  not  into  an  attic  with  a  ventilating  louver.  In  this  case  the  air 
may  ascend  one  flue  and  descend  another,  and  not  out  of  the  louver. 

The  quantity  of  air  taken  into  and  expired  from  the  lungs  by  a  single  indi- 
vidual is  quite  small,  probably  about  13  cubic  feet  on  an  average  per  hour. 
The  usual  gas-burner  delivers  from  4  to  6  cubic  feet  per  hour,  under  a  pressure 
of  1"  and  2"  of  water.  It  will  be  seen,  therefore,  how  small  apertures  are  neces- 
sary to  supply  the  lungs  of  a  person,  if  it  could  be  provided  directly  to  him 
and  taken  away  without  vitiating  other  air.  But,  in  addition,  air  is  vitiated  by 
personal  emanations,  and  consumed  by  lights.  These  last  can  readily  be  made, 
not  only  to  remove  all  their  products  of  combustion,  but  also  increase  the  cir- 
culation in  flues  for  the  ventilation  of  the  room. 

All  systems  of  ventilation  are  based  on  the  idea  that  so  many  individuals 
within  a  room  and  so  many  lights  burning  vitiate  so  much  air,  and  that  conse- 
quently a  very  large  quantity  of  outer  air  must  be  introduced  to  reduce  the  per- 
centage of  vitiation,  and  generally  with  very  little  consideration  as  to  the  distri- 
bution of  this  air,  although  it  is  in  every  one's  experience  that  the  air  in  some 
portions  may  be  fresh,  in  others  stifling  ;  that  in  hospital  wards  there  are  often 
dead  ends  where  the  air  does  not  circulate,  and  where  patients  do  not  as  a  rule 
recover.  The  system  is  to  provide  somewhere  in  a  room  air  enough,'  and  trust 
to  chance  for  its  distribution. 

Some  architects  make  the  educts  at  the  ceiling,  some  at  the  floor,  some  at 
both,  with  registers  to  control  the  openings.  For  sleeping-apartments,  if  there 
is  a  fireplace,  this  is  all  that  will  be  necessary  ;  if  the  air  goes  up  or  comes  down, 
it  does  not  make  draughts  about  the  heads  of  the  occupants. 

To   make   flues  draw,   various  forms  of 

T 1 r         chimney-tops  or   cowls   are   adopted.      The 

I /•"•"•"[*        k68^   and    simplest   are   the   Emerson   (Fig. 

MBE^          1227),  and  a  modification  of  the  same  (Fig. 

mlP      X          JK  \       1228)  ;  there  are  also  various  forms  of  self- 

lll — I  IIP 1 —         ncting  naPs>   turn-cowls,  etc.,  the  principle 

being  to  take  advantage  of  the  wind  to  make 
a  draught.     With  the  wind  blowing  across 

FIG.  1227.  FIG.  1228.  the  top  of  a  chimney,  a  bit  of  square-ended 

iron  pipe  extending  above  the  chimney  will 

answer  as  an  expirator,  but  without  a  wind  the  draught  must  depend  on  cir- 
cumstances within  the  dwelling  and  artificial  draught.  When  sufficient  cir- 
culation can  not  be  obtained  from  natural  differences  of  temperature  in  the 
atmosphere,  or  from  winds,  it  is  usual  to  have  recourse  to  fans,  to  force  air 
into  or  draw  it  from  a  building,  or  by  heat  applied  to  the  air  in  flues,  ducts,  or 
chambers  in  the  hot-air  furnaces.  Both  the  air  and  the  heat  are  necessary. 
When  heat  is  applied  for  ventilation  only,  as  in  mines,  a  fire  is  built  in  a  flue 
near  the  top,  and  the  air  necessary  for  combustion  is  drawn  from  the  mines  ; 
the  flue  extends  from  the  bottom  of  the  mine,  with  a  chimney  above  the  sur- 
face of  the  ground,  and  ducts  are  led  from  the  bottom  of  the  flue  to  the  face  of 
the  workings,  the  cold  air  for  ventilation  being  drawn  down  through  the  work- 
ing-shafts and  drifts.  In  buildings,  steam-pipes  and  gas-burners  are  put  in  flues. 


ARCHITECTURAL  DRAWING. 


549 


Methods  of  Heating. — The  open  fireplace  grate  heats  by  radiation,  commu- 
nicating heat  to  objects,  which  by  contact  transfer  it  to  the  air.  Persons  com- 
ing in  contact  with  rays  are  themselves  heated,  while  the  air  around  them  is 
cool  and  invigorating  for  breathing  ;  the  bright  glow  has  a  cheering  and  ani- 
mating effect  upon  the  system,  somewhat  like  that  of  sunlight.  As  a  ventilator, 
an  open  fire  is  one  of  the  most  important,  drawing  in  air  not  only  for  the  sup- 
port of  combustion,  but  also,  by  the  heat  of  the  fire  and  flue,  making  a  very 
considerable  current  through  the  throat  of  the  chimney  above  the  fire.  From 
this  cause,  although  there  is  a  constant  change  of  air,  yet  there  arises  one  great 
inconvenience  of  disagreeable  draughts,  especially  along  the  floor,  if  the  air- 
supply  be  drawn  directly  from  the  outer  cold  air  ;  but  in  connection  with  prop- 
erly regulated  furnaces  or  stoves,  the  open  fireplace  becomes  the  most  perfect 
means  of  heating  and  ventilation.  As  a  heater  merely,  the  open  grate,  in  very 
cold  weather,  is  not  satisfactory  ;  its  influence  is  only  felt  in  its  immediate 
vicinity,  and  but  from.  10  to  15  per  cent  of  the  heat  of  the  fuel  is  rendered 
available. 

Fig.  1229  represents  an  old  form  of  open  fire  used  in  a  tavern  bar-room  and 
office,  which  answered  admirably  for  heating  and  ventilation,  and  admitted 
of  access  to  many  persons.  It 
consisted  of  a  circular  grate  at  the 
level  of  the  floor  in  the  center  of 
the  room.  In  the  cellar  beneath 
was  an  ash-pit,  a,  in  brick-work, 
with  an  opening,  o,  to  supply  air  for 
the  combustion  of  the  fuel.  Above 
the  grate  was  a  counter-weighted 
sheet-iron  hood,  h,  connected  by  a 
pipe  with  the  chimney,  which  could 
be  raised  or  lowered,  to  suit  the  re- 
quired draught.  Around  the  grate 
was  a  ring-guard  to  rest  the  feet  on, 


and    the  customers  ranged  them- 
selves in  a  circle  round  the  fire. 

Stoves. — Open   stoves  heat   by  FIG.  1229. 

direct   radiation,    and   by  heating 

the  air  in  contact  with  them,  and  close  stoves  by  the  latter  way  only;  as 
economical  means  of  heating,  the  latter  are  the  best,  and,  when  properly 
arranged,  give  both  a  comfortable  and  wholesome  atmosphere.  There  should 
be  some  dish  of  water  upon  them  to  supply  a  constant  evaporation,  sufficient 
to  compensate  for  increased  capacity  of  the  air  for  moisture  due  to  its  in- 
creased heat.  In  the  hall  there  will  be  no  objection  to  a  close  stove,  letting 
it  draw  its  supply  of  air  as  it  best  can  ;  but  in  close  rooms  the  open  stove  is 
best,  on  the  plan  of  the  old  Franklin  stove,  or,  if  a  close  stove,  somewhat  on 
the  plan  of  a  furnace,  with  an  outer  air-supply  for  combustion  and  ventilation. 

Hot-air  furnaces  are  close  cast-iron  stoves,  inclosed  in  air-chambers  of  brick 
or  metal,  into  which  external  air  is  introduced,  heated,  and  distributed  by 
metal  pipes  to  the  different  rooms  of  a  house.  Furnaces  have  been,  of  late,  very 


550 


ARCHITECTUKAL   DRAWING. 


much  decried,  but  under  proper  regulation  they  are  very  cheap,  economical, 
and  even  healthful  means  of  ventilation  and  warming.  The  heating-surface 
should  be  very  large,  the  pot  thick,  or  even  incased  with  fire-brick,  that  it  may 
not  become  too  hot ;  there  should  be  a  plentiful  supply  of  water  in  the  cham- 
ber for  evaporation,  perhaps  also  beneath  the  opening  of  each  register  ;  the  air- 
supply  should  always  be  drawn  from  the  outer  air  and  unobjectionable  sources, 
through  ample  and  tight  ducts,  without  any  chance  of  draught  from  the  cel- 
lar ;  the  pot,  and  all  joints  in  the  radiator,  should  be  perfectly  gas-tight,  so 
that  nothing  may  escape  from  the  combustion  into  the  air-chamber.  With 
these  provisions  on  a  sufficient  scale,  and  proper  means  for  distribution  of 
the  heated  air  and  escape  of  foul  air,  almost  any  edifice  may  be  very  well 
heated  and  ventilated.  The  air  should  be  delivered  through  the  floor  or  the 
base-board  of  the  room,  and  at  the  opposite  side  from  the  flue  for  the  escape 
of  foul  air,  making  as  thorough  a  current  as  possible  across  the  room,  and 
putting  the  whole  air  in  motion.  In  dwelling-houses  the  fireplace  will  serve 
the  best  means  of  exit ;  in  public  rooms  distinct  flues  will  have  to  be  made 
for  this  purpose,  and  they  should  be  of  ample  dimensions  and  well  distrib- 
uted, with  openings  at  the  floor  and  ceiling  with  registers,  and  means  should 
be  provided  for  heating  the  flues.  An  architect,  in  laying  out  flues  for  heat- 
ing and  ventilation,  should,  both  in  plan  and  elevation,  fix  the  position  of 
hot  and  foul  air  flues,  and  trace  in  the  current  of  air,  always  keeping  in  mind 
that  the  tendency  of  hot  air  is  to  rise  ;  he  will  then  see  that,  if  the  exit- 
opening  be  directly  above  the  entrance- 
flue,  the  hot  air  will  pass  out,  warming 
the  room  but  little  ;  if  the  exit-opening 
be  across  the  room  and  near  the  ceiling, 
the  current  will  be  diagonal,  with  a  cold 
corner  beneath,  where  there  will  be  very 
little  circulation  or  warmth.  To  heat  the 
exit-flue,  a  very  simple  way  is  to  make  the 
furnace-flue  of  iron,  and  let  it  pass  up  cen- 
trally through  the  exit-flue. 

Fig.  1230  may  be  taken  as  a  type  of  a 
portable  (so  named  on  account  of  its  small 
size  and  metallic  case)  hot-air  furnace. 
The  air  is  introduced  at  the  bottom  of  the 
case,  passes  up  and  around  the  stove,  and 
out  through  the  ducts  D,  D,  D  to  different 
parts  of  the  building.  The  water-pan  p  is 
indispensable  to  the  hot-air  furnace,  and 
should  be  of  capacity  enough  for  a  day's 
supply,  or  have  automatic  means  of  keep- 
ing up  the  supply. 

Air  in  winter  is  very  dry,  but  as  its 
volume  is  enlarged  by  heat,  it  draws  a 
supply  of  moisture  from  everything  with  which  it  comes  in  contact — from  the 
skin  and  lungs,  creating  that  parched  and  feverish  condition  experienced  in 


FIG.  1230. 


ARCHITECTURAL  DRAWING.  551 

many  furnace-heated  houses  ;  from  furniture  and  wood-work,  snapping  joints 
and  making  unseemly  cracks. 

Thus,  taking  the  air  at  10°,  and  heating  it  to  70°,  the  ordinary  temperature 
of  our  rooms  requires  about  nine  times  the  moisture  contained  in  the  original 
external  atmosphere,  and,  if  heated  to  100°,  as  most  of  our  hot-air  furnaces 
heat  the  air,  it  would  require  about  23  times. 

The  portable  furnace  is  not  so  economical  as  the  furnace  set  in  brick -work, 
as  more  heat  escapes  through  the  metallic  case.  The  former  are  usually  made 
from  12"  to  24"  diameter  of  pot,  from  2'  to  4'  outside  diameter,  and  5'  to  6' 
height  of  case. 

The  brick-set  furnaces  are  from  20"  to  28"  pot,  outside  brick-work  from  5' 
to  6'  square,  walls  4"  thick,  height  6'  to  7'.  The  size  of  air-ducts  is  propor- 
tioned to  size  of  furnace.  The  inlet  should  be,  say,  equal  to  that  of  the  grate, 
and  the  sum  of  the  outlets  but  little  in  excess  of  this  area.  It  is  difficult  to 
give  any  rule  for  the  heating  capacity.  A  22"  pot  should  be  adequate  for  the 
heating  of  a  common  25'  x  60'  city  house,  and  the  higher  the  air-duct  the  less 
its  diameter. 

Steam  and  hot-water  circulation  are  applied  to  the  heating  of  buildings  by 
means  of  wrought  or  cast  iron  pipes  connected  with  boilers.  In  the  simplest 
form,  as  common  in  workshops  and  factories,  steam  is  made  to  give  warmth 
without  ventilation  by  direct  radiation  from  wrought-iron  pipes.  The  gen- 
eral arrangement  is  by  rows  of  1"  pipe  hung  against  the  walls  of  the  room,  or 
suspended  from  the  ceilings,  3'  of  1"  pipe  being  considered  adequate  to  heat  200 
cubic  feet  of  space  ;  if  there  are  many  windows  in  the  room,  or  the  building  is 
very  much  exposed,  more  length  should  be  allowed. 

Steam,  as  a  means  of  heating,  is  the  most  convenient  and  surest  in  its 
application  to  extensive  buildings  and  works.  From  boilers,  located  at  some 
central  point,  steam  can  be  conveyed  to  points  so  remote  that  in  many  cities  it 
is  matter  of  sale,  both  for  heating  and  power  purposes.  The  limits  of  the 
extension  of  steam-pipes  economically  have  not  yet  been  determined,  but  within 
the  range  of  the  buildings  occupied  by  any  single  textile  manufacturing  in- 
dustry steam-heating  has  proved  satisfactory,  and  is  of  almost  universal  adop- 
tion. For  stores,  warehouses,  large  buildings  of  all  sorts,  where  there  are 
extensive  or  numerous  rooms  to  be  heated,  steam  has  been  long  used,  and  the 
appliances  for  its  use  can  be  as  readily  obtained  in  all  our  cities  and  large  towns 
as  stoves  or  grates.  Steam  is  used  for  heating  at  either  high  or  low  pressures  ; 
under  5  or  6  pounds  would  be  considered  low  pressure.  A  low-pressure  ap- 
paratus may  draw  direct  from  a  boiler,  or  be  supplied  from  the  exhaust  of  a 
steam-engine  ;  if  from  the  latter,  a  certain  amount  of  back  pressure  must  be 
put  on  the  engine  to  establish  a  circulation  in  the  steam-heating  pipes. 

In  the  operation  of  heating  by  steam,  the  steam,  in  giving  off  its  latent 
heat  through  the  pipes  to  the  air  of  the  room,  returns  to  water ;  the  apparatus 
would  then  be  nothing  but  pipes  to  convey  the  steam  to  radiators  to  condense 
it,  and  pipes  to  return  the  water  to  the  boiler,  were  it  not  for  air  invariably  in 
water  and  steam.  This  necessitates  a  more  complicated  circulation  ;  there  should 
be  a  regular  flow  outward  of  steam  from  the  boiler,  and  inward  of  water  and 
steam  to  it,  both  as  far  as  possible  together,  and  in  the  same  direction.  When 


552 


ARCHITECTURAL  DRAWING. 


hot  water  is  used  for  heating,  there  must  be  circulation  throughout  the  sys- 
tem ;  the  water  flows  out  from  the  top  of  the  boiler,  gives  out  its  heat,  and 
returns,  practically  of  the  same  bulk,  cold  to  the  bottom  of  the  boiler,  and 
any  radiator  out  of  the  line  of  this  current  is  of  no  use.  A  single  valve  shuts 
off  the  circulation  in  the  hot-water  apparatus,  while  two  are  necessary  with  a 
steam  apparatus,  for  the  steam  cut  off  on  the  direct  pipes  may  back  up  through 
the  return-pipe. 

Steam  is  used  for  heating  rooms  either  directly  or  indirectly.  Direct  steam- 
heating  is  like  that  of  common  stoves,  without  any  considerations  for  ventila- 
tion. Indirect  steam-heating  is  like  that  of  hot-air  furnaces.  Steam  radiators 
are  inclosed  in  a  box  or  chamber,  into  which  air  is  drawn  or  forced,  and  then 
distributed  by  ducts  to  the  rooms  to  be  warmed  and  ventilated.  Thus,  when 
ventilation  is  combined  with  steam  or  hot-water  heating,  the  metallic  surfaces 
brought  in  contact  with  the  air  usually  range  from  212°  to  250°,  while  the  pot 


4,0. 


FIG.  1231. 


of  the  air-furnace  may  be  near  a  white  heat.  In  a  sanitary  point  of  view,  hot- 
water  or  low-steam  coils  in  air-chambers  are  a  more  surely  healthy  means 
of  warming  and  ventilation  ;  the  greatest  objection  is  their  expense,  the  care 
requisite  in  attending  them,  and  the  danger  of  freezing  and  bursting  the  pipes 


ARCHITECTURAL  DRAWING. 


553 


if  worked  intermittently  in  winter.  In  the  arrangement  it  is  usual,  in  dwell- 
ing-houses, to  place  the  coils  at  different  points  in  the  cellar,  as  near  as  possible 
beneath  the  rooms  to  be  heated.  In  public  buildings  frequently  a  very  large 
space  in  the  cellar  is  occupied  by  the  coils,  into  which  the  air  is  forced  by  a 
fan,  and  then  distributed  by  flues  or  ducts  throughout  the  building. 

All  inlet  or  outlet  ventilating  flues  should  be  provided  with  dampers  or 
registers,  to  control  the  supply  or  discharge  of  air,  cutting  it  off  when  sufficient 
heat  is  secured,  or  retaining  the  warmth  when  ventilation  is  not  required. 

Fig.  1231  (an  illustration  from  "The  Sanitary  Engineer")  is  the  plan  of  a 
portion  of  a  large  building  heated  by  steam.  B  B  are  two  boilers,  either  of 
which  would  be  sufficient  for  the  purpose  ;  the  steam  mains  are  shown  by  black 
lines  following  those  of  the  building,  with  the  sizes  marked  upon  them ;  the 
risers  by  inclined  lines,  with  the  square  foot  of  radiating  surface  on  each  story, 
marked.  This  is  a  very  convenient  form  of  drawing,  explanatory  of  the  sys- 
tem. It  is  usual  to  draw  the  steam  mains  and  risers  in  red,  and  the  returns  in 
black,  with  the  diameters  on  each. 

Fig.  1232  is  the  elevation  of 
a  small  steam-heating  apparatus, 
illustrating  the  general  action.  B 
is  the  boiler,  and  R  and  R'  radia- 
tors on  different  stories  ;  s  is  the 
steam-pipe,  and  r  r'  return  or  drip 
pipes.  The  steam  is  drawn  from 
the  top  of  the  boiler,  and  the  re- 
turns must  be  below  the  surface, 
W.  L.,  of  the  water  in  the  boiler. 
The  circulation  is  simple  and  in- 
telligible, and  applicable  to  a  hot- 
water  apparatus ;  as  a  steam  ap- 
paratus, if  it  is  required  to  shut 
•off  the  lower  radiator,  R,  both 
the  inlet  and  outlet  valves  on  the 
radiator  must  be  shut.  If  only 
the  top  valve  be  shut,  the  steam 
in  the  radiator  will  be  condensed, 
and  the  pressure  from  the  boiler 
will  fill  it  with  water.  If  the  lower 
valve  only  be  shut,  the  radiator 
will  still  act  as  a  condenser  till  it 
is  filled  with  water.  In  the  upper 
radiator,  R',  there  is  no  outlet- 
valve,  as  the  radiator  is  supposed 
to  be  set  at  a  level  above  the 
height  to  which  the  water  would  be  raised  by  the  pressure  of  the  steam  in  the 
boiler.  This  arrangement  of  separate  returns  for  each  radiator  is  sometimes 
used,  but  the  usual  practice  is  to  have  single  returns,  into  which  there  are 
branches  from  each  radiator,  controlled  by  valves.  In  low-steam  apparatus, 


FIG.  1232. 


554 


ARCHITECTURAL   DRAWING. 


the  steam  is  introduced  and  the  water  removed  by  the  same  pipe,  and  con- 
trolled by  a  single  valve. 

Fig.  1233  is  an  elevation,  showing  the  usual  arrangement  of  mains,  s  s,  and 

returns,  rr,  when  the  hori- 
zontal distance  from  the  boiler 
is  small  and  the  risers  few. 
The  inclination  of  the  mains 
is  toward  the  boiler,  and  their 
condensed  water  returns  by 
them  to  the  boiler. 

Fig.  1234  is  the  better 
practice,  and  necessary  if  the 
steam  is  high  pressure,  the 

mains  extended,  and  the  branches  numerous.  The  inclination  of  the  mains,  s  s, 
is  from  the  boiler,  and  the  condensed  water  flows  down  to  the  lowest  angle, 
where  it  is  connected  with  the  return,  r,  and  is  by  this  brought  back  to  the  boiler. 
The  size  of  the  boiler  for  a  steam-heating  apparatus  is  based  on  the  amount 
of  radiating  surface,  which  must  include  that  of  the  steam-mains,  if  not 
clothed,  and  of  the  returns.  But,  as  boilers  vary  so  much  in  their  proper- 


FIG.  1233. 


FIG.  1234. 


tions,  it  is  impossible  to  give  a  rule  applicable  to  all  of  them.  Some  estimate  by 
boiler-grate  surface,  500  square  feet  of  radiating  surface  to  each  square  foot  of 
grate  ;  some,  1  H.  P.  of  boiler  to  each  200  square  feet  of  radiating  surface. 
The  amount  of  radiating  surface  depends  on  the  cubic  feet  of  air  to  be  heated. 
It  is  usual  to  estimate  that  from  150  to  200  cubic  feet  of  room-space  can  be 
heated  from  0  to  70°  by  1  square  foot  of  radiating  surface  ;  or,  say,  4  run- 
ning feet  of  f "  pipe  or  3  feet  of  I"  pipe.  But  this  is  to  be  modified  very  much 
by  the  exposure  of  the  room,  the  amount  of  glass  surface,  the  thickness  of 
wall,  and  the  temperature  of  surroundings.  The  effect  of  glass  as  a  cooling 
surface  can  be  readily  understood  by  the  difference  one  experiences  in  the  heat 
of  cars  in  motion  or  stopped,  and  the  advantages  of  double  windows  in  the 
same  conveyances. 

Where  the  heating  is  indirect,  as  there  are  more  cubic  feet  of  air  to  be  heat- 
ed, the  radiating  surface  is  to  be  increased,  usually  to  about  three  times  that  of 
the  direct  heating. 

C.  B.  Eichards  says  that,  for  direct  radiators,  1  square  foot  of  surface  gives 
off  3  heat-units  for  each  degree  (1°)  difference  of  temperature  between  the  air 
of  the  room  and  that  of  the  steam  in  the  radiator. 

As  the  boiler  must  be  proportioned  to  the  requirements  of  heating,  as  de- 
termined by  the  square  feet  of  radiators,  the  sizes  of  mains  and  returns  are 
also  measured  by  the  same  standard.  A  common  rule  is  I"  diameter  of  main 


AKCHITECTUEAL  DRAWING.  555 

for  each  100  square  feet  of  radiating  surface,  varying  with  the  squares  of  the 
diameters— 2",  400  square  feet;  3",  900  ;  4",  1,600.  In  fact,  the  larger  sizes 
will  be  sufficient  for  a  much  larger  radiating  surface  than  given  by  this  rule. 

For  the  returns,  one  size  less  than  that  of  the  steam  mains  is  the  rule  ; 
thus,  a  f "  return  for  a  1"  pipe,  but  no  pipe  of  less  diameter  than  f "  is  used  ; 
for  a  2£"  steam  a  2"  return,  and  a  larger  than  2"  is  seldom  used.  It  may  not 
be  always  practicable  to  return  the  condensed  water,  as  shown  in  the  figures 
above,  by  gravitation,  but  there  are  various  forms  of  receivers  or  traps  in  which 
the  water  is  collected  and  returned,  automatically  as  in  the  Albany  trap,  or  by 
pumping  to  the  boiler. 

Figs.  1235  to  1241  are  common  forms  of  radiators.  Fig.  1235  is  a  bench 
coil,  often  called  a  mitre  coil,  from  the  vertical  or  horizontal  angle  made  at  the 
end  of  the  pipes  to  admit  of  their  unequal  expansion.  In  the  circulating  coil 
(Fig.  1236),  often  called  the  trombone,  the  circulation  is  alternately  forward 
and  back ;  when  placed  in  rows,  as  in  Fig.  1237,  it  is  a  box  coil ;  the  ends  of 
the  pipes  at  both  top  and  bottom  are  connected  in  heads.  Fig.  1238  is  a  hori- 
zontal radiator,  similar  in  its  action.  Figs.  1239  and  1240  are  vertical  radia- 
tors, the  first  composed  of  wrought-iron  pipes  inserted  in  a  hollow  cast-iron 
base,  circulation  being  obtained  by  a  sheet-iron  division  in  the  pipe  as  in  the 
Nason  radiator,  by  an  inside  pipe,  or  by  the  connection  of  two  pipes  at  top  by 
a  return  bend.  Fig.  1240  is  a  Bundy  radiator,  in  which  there  are  twin  cast- 
iron  pipes  connected  at  the  top  and  bottom.  Fig.  1241  are  cast-iron  pin  radia- 
tors, so  called  from  the  projections,  effective  for  indirect  radiators.  In  meas- 
uring the  surface  of  circulating  coils,  include  the  lengths  of  angles  and  all 
fittings  ;  in  the  vertical  radiators,  include  the  base. 

On  the  radiator  (Fig.  1240)  a  small  pipe  (p)  will  be  seen,  which  is  an  air- 
vent,  often  automatic,  but  indispensable  for  a  prompt  start  of  the  circulation. 

Plumbing. — The  conveniences  for  comfort  in  modern  buildings  require  the 
introduction  of  water  and  its  removal.  Most  cities  have  water-supplies  and  a 
system  of  sewers,  and  the  plumber  makes  the  connections  with  both.  In  the 
country,  for  the  better  class  of  houses  there  are  private  expedients  to  supply 
their  places,  largely  by  wells  and  pumping,  and  connections  to  cesspools.  The 
quantity  used  in  each  household  varies  with  the  wants  and  habits  of  the  occu- 
pants. An  average  bath  will  take  25  gallons  ;  each  use  of  a  water-closet  from 
2  to  3  gallons.  A  wash-tub  will  hold  from  10  to  20  gallons.  If  the  water  is  to 
be  pumped  by  hand,  from  7  to  10  gallons  will  be  reckoned  as  the  use  by  each 
person ;  if  from  aqueduct,  30  to  50  gallons  is  ample. 

The  regulation  size  of  taps  for  city  mains  is  from  £"  to  -f-",  and  the  pipes 
leading  into  the  house  from  f "  to  1"  diameter.  The  pipes  are  usually  of  lead, 
as  most  waters  are  not  affected  sensibly  by  lead,  if  the  pipes  are  always  kept 
full,  and  there  is  fair  circulation.  In  some  cases  block-tin  pipes  are  used  ;  or 
iron,  galvanized,  or  coated  with  some  preparation  of  asphalt,  or  glass-lined. 

The  soil  or  house-sewer  pipe  connections  with  the  main  sewer  or  cesspool 
are  usually  vitrified  stone-ware  pipe,  from  4"  to  6"  diameter,  as  they  are  not 
only  for  the  discharge  of  the  sewage,  but  also  for  the  rainfall  from  the  roof. 
Within  the  house  the  pipe  is  either  of  stone-ware  or  cast-iron  ;  invariably  of 
the  latter  if  the  pipe  is  exposed.  The  rising  pipe  to  the  roof  is  here,  also, 


556 


ARCHITECTURAL  DRAWING. 


FIG.  1241. 


AKCHITECTURAL  DRAWING. 


557 


FIG.  1242. 

usually  of  cast-iron,  and  4"  diameter  may  be  considered 
ample  for  a  common  house  ;  the  smaller  branches  may 
also  be  of  iron,  but  when  as  small  as  2"  are  usually  of  lead. 

Fig.  1242  is  the  perspective  of  a  kitchen-range  boiler 
and  sink  :  c  is  the  cold-water  pipe  leading  to  the  sink  and 
to  the  boiler ;  it  enters  the  top  of  the  boiler,  and  is  led 
down  nearly  to  the  bottom.  The  hot  water  is  drawn  from 
the  top,  through  the  pipe  h,  is  led  down  to  the  sink  and 
up  for  distribution  through  the  house.  The  water  is  heat- 
ed in  the  boiler  by  the  connection  with  the  water-back  of 
the  range,  r ;  the  water  flows  through  the  pipe,  I,  is  con- 
nected with  the  lower  part  of  the  water-back,  and  returns 
by  the  pipe,  u,  from  the  top  of  the  water-back  to  a  higher 
point  in  the  boiler  ;  b  is  the  blow-off  pipe. 

It  will  be  observed  that  at  the  draw-cocks  over  the 
sink  there  are  pipes,  a  a,  turned  up  ;  these  are  air-cham- 
bers, to  cushion  the  blow  of  the  water-hammer  when  the 
cocks  are  shut  quickly.  Beneath  the  sink  there  is  a 
trapped  connection  with  the  sewer-pipe. 

Fig.  1243  is  the  elevation  of  a  gal vani zed-iron  boiler, 
but  those  in  general  use  here  are  of  copper. 

Fig.  1244  is  the  perspective  drawing  of  a  cast-iron 
sink,  of  the  usual  form  and  material.  They  are  to  be 
obtained  of  all  suitable  dimensions,  rectangular,  from  16"  FIG 


658 


ARCHITECTURAL  DRAWING. 


X  12"  X  5"  deep,  to  96"  X  24" 
X  10"  deep ;  also,  half -circle 
and  corner  sinks,  and  deep  and 
slop  sinks. 

In  the  kitchen,  or  a  laun- 
dry-room adjacent,  tubs  are 
set  for  washing,  with  hot  and 
cold  water  service.  The  water- 
pipe  connections  are  usually 

}",  the  waste  connections  2".  The  tubs  themselves  are  mostly  of  wood,  but 
there  are  many  of  cast-iron  (Fig.  1245),  galvanized  or  enameled,  of  slate,  and 
of  earthenware. 


FIG.  1244. 


FIG.  1245. 

In  the  butler's  pantry  there  is  usually  a  sink  set  of  planished  tinned-copper, 
with  hot  and  cold  water  connections. 

In  the  chambers  and  dressing-rooms,  bowls  of  earthenware  are  set,  with  like 

connections.  The  sizes  of 
basins  vary  from  12"  to 
18"  outside  diameters. 

Fig.  1246  shows  the 
usual  form  of  setting  of  a 
wash-basin  in  a  counter- 
sunk marble  slab,  with  a 
back  of  the  same  mate- 
rial ;  these  are  the  com- 

FIG.  124«.  mon  ground  key  swinging 

faucets  for  the  supply  of 

hot  and  cold  water,  and  the  waste  is  closed  by  a  metal  or  rubber  plug,  at- 
tached to  a  chain,  with  the  other  end  fastened  to  a  pin  in  the  marble  slab. 


ARCHITECTURAL  DRAWING.  559 

The  sides  are  inclosed  with  wood,  forming  a  closet  beneath  the  basin,  with 
usually  small  drawers  for  towels  at  each  side  of  the  closet. 

Fig.  1247  is  a  cast-iron  bath-tub,  of  a  simple  pattern,  with  an  overflow,  o, 
and  apertures  c  and  h  near  the  bottom  for  hot  and  cold  water  connections  ;  the 
waste  is  closed,  as  in  the  basin  above,  by  a  plug.  When  there  is  no  overflow  to 
the  tub,  this  plug  is  a  hollow  pipe,  down  which  there  is  an  overflow  when  its 
lower  extremity  or  annular  plug  closes  the  waste-pipe.  Bath-tubs  are  more 
generally  made  of  planished  tinned-copper  in  a  wooden  box  or  support,  and 
inclosed  by  wooden  panels.  The  more  expensive  bath-tubs  are  made  of  porce- 
lain, and  may  or  may  not  be  inclosed.  In  most  bath-rooms  there  is  a  set  wash- 
hand  basin  and  a  water-closet — often  a  foot-bath  and  bidet-pan.  Formerly  it 
was  the  common  practice  to  have  but  one  trap  beneath  the  water-closet,  into 
which  all  the  waste-pipes  discharged,  but  of  late  the  water-closet  connection 
with  the  soil-pipe  is  independent  of  the  others. 


FIG.  1247. 

It  is  preferable  to  make  the  water-closet  in  a  separate  room,  distinct,  with 
Its  own  water  and  sewer-service  and  means  of  ventilation. 

The  construction  of  one  form  of  water-closet,  with  all  the  modern  appli- 
ances for  the  removal  of  soil  and  for  ventilation,  will  be  understood  from  the 
section  (Fig.  1248).  The  seat  is  not  shown,  but  is  just  above  the  basin,  B, 
which  contains  some  water  to  receive  the  defecations,  to  prevent  the  soil  attach- 
ing to  the  side  of  the  basin,  and  in  a  measure  to  check  its  offensive  smell.  T 
is  the  trap  or  water-seal  which  prevents  the  smell  from  the  soil-pipe  S  passing 
up  through  the  basin.  The  water-discharge  from  the  pipe  W  is  through  a  rim- 
flush  around  the  edge  of  the  basin.  ^The  sudden  discharge  washes  out  the  basin 
B  into  the  trap  T,  which  is  also  cleaned  by  the  rush  of  water.  The  soil-pipe  S 
extends  up  through  the  roof,  and  may  or  may  not  also  serve  as  a  rain-leader. 
A  sudden  flow  of  water  down  the  soil-pipe  often  acts  as  an  ejector  to  draw  the 
water  out  of  the  trap  T,  and  break  the  water-seal ;  to  prevent  this,  there  is 
an  air  connection,  A,  leading  also  to  the  top  of  the  house.  But  as  the  offense 
of  a  water-closet  is  largely  due  to  its  recent  use,  and  as  smell  once  getting  into 


560 


ARCHITECTUKAL  DRAWING. 


the  room  is  with  difficulty  and  slowly  removed,  there  is  a  ventilating-pipe,  V, 
connecting  the  basin  B  with  a  ventilating-flue.  It  will  be  observed  that  this  is 
the  most  important  part  of  the  apparatus ;  connected  with  a  chamber  commode, 
it  would  remove  all  smell,  and  if  there  were  no  trap  to  the  soil-pipe,  or  were 


w 


FIG.  1248. 

the  water-seal  broken,  it  would  still  prevent  any  offensive  smell  from  pene- 
trating the  house.  If  the  soil-pipe  be  made  also  a  ventilating-pipe,  as  is  fre- 
quently done  by  its  connection  with  the  hot-air  flue,  then  the  trap  and  pipes 

A  and  V  are  unnecessary. 

Fig.  1249  is  an  elevation  of  the  simplest 
form  of  closet — the  hopper-closet — and  in  many 
respects  the  best.  It  is  shown  in  section  (Fig. 
1250),  with  its  water,  soil-pipe,  and  water  con- 
nection. By  pulling  up  the  handle,  A,  the  disk- 
valve  in  the  cistern  is  raised,  and  water  sup- 
plied to  the  closet-basin. 

Fig.  1251  is  the  section  of  a  pan-closet,  for 
many  years  the  most  popular  closet.  The  cop- 
per pan,  when  shut,  cuts  off  the  view  of  the 
trap  below  and  any  odor  from  it ;  with  a  small 
flow  of  water  the  basin  is  readily  kept  clean,  but  soil  is  apt  to  lodge  in  the  iron 
receiver,  and  the  odor  to  arise  from  it  when  the  pan  is  down.  There  is  an 


FIG.  1249. 


ARCHITECTURAL  DRAWING. 


561 


annular  ventilating-tube  beneath  the  seat,  with  an  air-shaft  attached,  but  of 
altogether  inadequate  dimension  for  the  purpose,  as  may  be  said  of  all  such 
vents  attached  to  water-closets.  There  is  also  the  air- vent  to  prevent  the  water 
being  drawn  from  the  trap.  No  water  connections  are  shown  in  the  figure. 


SEAT 


FLOOR 


LEAD  TRA 


FIG.  1251. 


Fig.  1252  is  the  section  of  a  flap- 
closet,  in  which  a  flap-valve  supplies 
the  place  of  a  pan. 


FIG.  1250. 


FIG.  1252. 


JET  CLOSET. 


Fig.  1253  is  the  section  of  a  siphon-jet  closet.     In  addition  to  the  fan 
flush,  /,  into  the  basin,  it  has  a  jet-pipe  j  at  its  bottom,  inducing  a  current  in 
the  direction  of  the  inclined  leg  of  the 
trap,  and  by  flush  and  jet  the  water  is 
siphoned  from  the  basin. 

The  use  of  traps  has  already  been  ex- 
plained, but  they  are  varied  in  their  form, 
all  answering  the  same  purpose,  to  cut  off 
the  air-connection  of  the  soil-pipe  with 
the  room  in  which  the  appliance  is 
placed.  The  smaller  traps  are  invariably 
in  lead. 

36 


FIG.  1253. 


ARCHITECTURAL  DRAWING. 


Figs.  1254  to  1261  represent  the  usual  forms  of  lead  traps.  It  will  be  ob- 
served that  there  are  screw-plugs  at  the  bottom  of  the  traps,  which  can  be  taken 
out  to  remove  any  obstruction.  As  the  water  may  be  drawn  out  of  any  trap 
by  the  passage  of  water  down  the  pipe  with  which  it  is  connected,  an  air-vent, 
as  already  described,  in  the  water-closet  trap,  is  put  on  these  small  traps.  In- 
stead of  this,  by  inserting  the  rising  pipe  at  #,  so  that  water  from  the  waste 
above  should  drip  a  little  into  the  lower  trap,  draft  from  it  is  prevented. 


9*8. 


SHORT  BEND.   LONG  BEND. 


li_P 

FIG.  1254.   FIG.  1255.  FIG.  1256.   FIG.  1257.   FIG.  1258.    FIG.  1259.   FIG.  1260.    FIG.  1261. 


Figs.  1262  and  1263  are  cast-iron  traps,  with  a  cap  that  may  be  removed  to 
clean  the  trap,  or  the  aperture  may  be  used  for  air-vent  connection. 


S-TRAP.  TRAP  WITH  SIDE  OUTLET. 


FIG.  1262. 


FIG.  1263. 


FIG.  1264. 


FIG.  1265. 


Fig.  1264  is  the  section  of  a  foZ?-trap,  used  on  sinks,  with  a  strainer,  S,  above  it. 
Fig.  1265  is  a  plate  with  plug,  for  the  bottom  of  sinks  and  bath-tubs. 
Figs.  1266  to  1271  are  common  cast-iron  bends  or  angles. 


QUARTER 
BEND. 


DOUBLE  HUB, 
QUARTER  BEND. 


EIGHTH 
BEND. 


SIXTH 
BEND. 


SIXTEENTH 
BEND. 


KETURN 
BEND. 


FIG.  1268. 


FIG.  1269. 


FIG.  1270.        FIG.  1271. 


HALF 
Y-BRANCH.        Y-BRANCH. 


DOUBLE 

Y-BRANCH. 


DOUBLE  HALF 
Y-BRANCH. 


FIG.  1272. 


FIG.  1273. 


FIG.  1274.         FIG.  1275. 


FIG.  127*. 


FIG.  1277. 


Figs.  1272  to  1277  are  cast-iron  branches.     The  T  branch  and  cross-head 
are  objectionable,  as  the  flows  from  the  branches  and  mains  are  at  right  angles. 


ARCHITECTURAL  DRAWI 


563 


and  mutually  obstructive ;  whereas  in  the  Y,  especially^6  full  Y,  the  flows 
are  at  acute  angles  with  each  other,  and  the  currents  con?e£ggr>  Similar  fit- 
tings are  used  for  water,  but  they  are  much 
heavier. 

Most  water-closet  basins  are  inclosed  by 
a  lidded  seat  and  riser,  but  the  less  wood- 
work about  a  basin  the  better.  The  seat  is 
generally  hung  with  hinges,  so  that  it  can 
be  raised,  and  the  basin  used  as  a  urinal  for 
men  ;  the  upper  edge  of  the  basin  being  ex- 
tended or  covered  with  an  earthenware  tray, 
sloping  toward  the  basin. 

Urinals,  of  which  one  form  is  shown 
(Fig.  1278),  are  often  used  in  public  build- 
ings, and  in  airy  situations ;  although  they 
have  water  connection,  w,  and  a  rim  flush, 
it  is  almost  impossible  to  keep  them  sweet ; 
a  cake  of  carbolic  soap  is  often  put  in  the 
basin,  but  the  most  effectual  means  adopted 
on  many  railway-cars  is  a  piece  of  ice.  As 
the  raising  of  the  seat  of  the  water-closet 

makes  this  convenience  a  good  urinal,  the  distinctive  one  is  but  little  used  in 
private  houses. 

The  Water- Service  to  Water- Closets. — In  the  cheaper  hoppers  the  supply  is 
often  directly  from  the  house- 
service.  In  these  the  trap  is 
well  down,  and  the  flush,  if 
not  certain,  there  is  nothing 
B 


FIG.  1278. 


FIG.  1279. 

objectionable  to  sight.  The 
water  may  be  let  on  by  hand, 
or  by  an  automatic  valve  con- 
nected with  sitting  down  on 
the  seat,  or  by  opening  or  shut-  FIG.  1280. 

ting  the  closet-door. 

As  the  supply  from  the  service  is  uncertain  if  there  is  a  draught  in  an- 
other quarter,  it  is  now  more  common  to  have  a  cistern-supply,  shown  in 


561  ARCHITECTURAL  DRAWING. 

section,  Fig.  1279.  B  is  a  ball-cock,  operating  a  valve  in  the  water-pipe,  by 
which  the  water  is  admitted  to  the  cistern  whenever  the  water  is  below  a  cer- 
tain level ;  /  is  a  lever,  by  which  the  discharge- valve  is  raised  or  lowered  ;  the 
valve-opening  is  large  and  the  water  flows  into  a  service-box,  S,  filling  it,  and 
at  the  same  time  discharging  through  the  p  into  the  closet-basin.  When  the 
valve  is  closed,  the  water  still  continues  to  flow  from  the  service-box,  vent  being 
given  through  the  air-pipe,  which  in  this  case  serves  also  as  an  overflow. 

Fig.  1280  is  the  section  of  another  cistern,  in  which  the  ball,  B,  or  float, 
operates  a  common  plug-valve,  A.  The  service-box,  D,  acts  as  a  sort  of  a 
measure  of  the  quantity  of  water  used.  When  it  is  filled  by  means  of  the 
valve  G,  the  valve  H  is  closed,  and  then,  when  the  valve  H  is  raised  for  the 
flush  of  the  closet,  G  is  closed.  There  is  an  air- vent  around  the  chain  or  rod 
of  the  valve  H,  and  the  overflow  E  is  independent.  The  supply-pipe  is  ex- 
tended nearly  to  the  bottom  of  the  cistern  by  a  short,  loose  pipe,  as  shown  at 
L,  to  avoid  noise  from  falling  water. 

Lighting. — It  may  be  needless  to  say  that  the  light  in  a  building  should  be 
as  much  as  possible  from  natural  sources,  as  it  conduces  to  health  and  cleanli- 
ness, and  economy  in  conducting  any  industrial  pursuits.  But,  for  artificial 
lighting,  the  present  permanent  fixtures  are  usually  for  the  use  of  gas.  In  the 
distribution  of  gas  through  the  building,  wrought-iron  pipes  are  invariably 
used.  The  old  English  rule  for  the  sizes  of  these  pipes  : 

i" 6  feet  long 1  outlet.  1"   70  feet  long 35  outlets. 

f" 20        "        3  outlets.  1£" 100         "        60       " 

i" 30        "       6      "  1-J" 150         "        100       " 

f" 90        "        20      "  2" 200         "        200       " 

The  couplings  to  elbows  are  similar  to  those  used  in  steam-fitting,  but 
lighter ;  'the  cocks  are  the  common  plug-cocks. 

Gas  fittings  are  in  all  forms  of  brackets  and  pendants,  with  any  number  of 
branches,  with  fixed,  swing,  and  slide  joints,  and  burners  in  great  variety. 
Bat-wing  and  fish-tail  tips  and  Argand  burners  are  the  most  used,  with  or 
without  globes  and  shades.  The  Argand  must  have  a  chimney.  Consump- 
tion of  gas  is  commonly  from  3  to  6  ft.  per  hour  per  burner. 

GREEK   AND    ROMAN   ORDERS   OF   ARCHITECTURE. 

In  themselves,  and  for  the  purposes  of  construction,  the  "  orders  of  archi- 
tecture "  are  now  of  little  utility ;  but,  as  examples  of  proportions  of  graceful 
curves  and  outlines,  they  are  useful  as  studies  and  manual  practice  for  the 
draughtsman. 

The  Tuscan,  Doric,  Ionic,  Corinthian,  and  Composite  orders,  are  systems 
or  assemblages  of  parts  subject  to  certain  uniform  established  proportions, 
regulated  by  the  office  each  part  has  to  perform,  consisting  of  two  essential 
parts,  a  column  and  entablature,  subdivided  into  three  parts  each  :  the  first 
into  the  base,  the  shaft,  and  the  capital ;  the  second  into  the  architrave,  or 
chief  beam,  0,  Fig.  1281,  which  stands  immediately  on  the  column  ;  the 
frieze,  B,  which  lies  on  the  architrave  ;  and  the  cornice,  A,  which  is  the 
crowning  or  uppermost  member  of  an  order.  In  the  subdivisions  certain 


* 


ARCHITECTURAL  DRAWING. 


f-l                                                                i 

r 

c 

;*--  —  -2z%—  -» 

r 

FIG. 


1281. 


FIG.  i  1282. 


fi 


FiG.h283. 


565 


*$ 


2tfl/ 


.  1284.J 


FIG.  1285. 


566  ARCHITECTURAL   DRAWING. 

horizontal  members  or  moldings  are  used :  thus,  the  ogee  (a),  the  corona  (b), 
the  ovolo  (c),  the  cavetto  (d),  with  the  fillets,  compose  the  cornice  ;  the  fasciae 
(ff)>  the  architrave  ;  the  abacus  (g),  the  ovolo  (c),  the  astragal  (ii),  and  the 
neck  (h),  are  the  capital  of  the  column  ;  the  torus  (k)  and  the  plinth  (/)  (Fig. 
1283)  are  the  base.  The  character  of  an  order  is  displayed,  not  only  in  its 
column,  but  in  its  general  forms  and  details,  whereof  the  column  is,  as  it  were, 
the  regulator ;  the  expression  being  of  strength,  grace,  elegance,  lightness,  or 
richness.  Though  a  building  be  without  columns,  it  is  nevertheless  said  to 
be  of  an  order,  if  its  details  be  regulated  according  to  the  method  prescribed 
for  such  order. 

In  all  the  orders  a  similar  unit  of  reference  is  adopted  for  the  construction 
of  their  various  parts.  Thus,  the  lower  diameter  of  the  column  is  taken  as 
the  proportional  measure  for  all  other  parts  and  members,  for  which  purpose 
it  is  subdivided  into  sixty  parts,  called  minutes,  or  into  two  modules  of  thirty 
minutes  each.  Being  proportional  measures,  modules  and  minutes  are  not 
fixed  ones  like  feet  and  inches,  but  are  variable  as  to  the  actual  dimensions 
which  they  express — larger  or  smaller,  according  to  the  actual  size  of  the 
diameter  of  the  column.  For  instance,  if  the  diameter  be  just  five  feet,  a 
minute,  being  one  sixtieth,  will  be  exactly  one  inch. 

To  draw  an  elevation  of  any  one  of  the  orders,  determine  the  diameter  of 
the  column,  and  from  that  form  a  scale  of  equal  parts  by  sixty  divisions,  and 
then  lay  off  the  widths  and  heights  of  the  different  members  according  to  the 
proportions  of  the  required  order,  as  marked  in  the  body  or  on  the  sides  of  the 
figures. 

Figs.  1281  to  1285  are  illustrations  of  the  Tuscan  order  :  e,  in  the  frieze 
corresponding  to  the  Doric  triglyph,  may  or  may  not  be  introduced.  Fig. 
1281  is  an  elevation  of  the  capital  and  entablature ;  Fig.  1283  of  the  base ; 
and  Fig.  1282  of  another  capital. 

A  slightly  convex  curvature,  or  entasis,  is  given  in  execution  to  the  outline 
of  the  shaft  of  a  column,  by  classic  architects,  to  counteract  a  fancied  appear- 
ance of  concave  curvature,  which  might  cause  the  middle  of  the  shaft  to  ap- 
pear thinner  than  it  really  is. 

Fig.  1284  represents  the  form  of  a  half-column  from  the  Pantheon  at 
Rome.  In  Fig.  1285,  another  example,  the  lower  third  of  the  shaft  is  uni- 
formly cylindrical.  The  entasis  of  the'  two  thirds  is  constructed  by  dividing 
the  arc,  a  ft,  into  equal  parts,  and  the  columns  into  the  same  number,  and  pro- 
jecting the  divisions  of  the  arc  on  to  those  of  the  column.  The  upper  diame- 
ter of  column  or  chord  at  b  is  52  minutes. 

Figs.  1286  to  1290  exhibit  an  example  of  the  Doric  order,  from  the  Temple 
of  Minerva,  in  the  Island  of  Egina.  Fig.  1286  is  an  elevation  of  the  capital 
and  the  entablature ;  Fig.  1287  of  the  base,  and  a  part  of  the  podium ;  Fig. 
1288  shows  the  forms  of  the  flutes  at  the  top  of  the  shaft,  and  Fig.  1289  at 
the  base  ;  Fig.  1290  the  outline  of  the  capital  on  an  enlarged  scale. 

The  mutules,  a  a,  the  triglyphs,  b  b,  the  guttae  or  drops,  d  d,  of  the  entabla- 
ture, the  echinus,/,  and  the  annulets,  g  g,  of  the  capital,  may  be  considered 
characteristic  of  the  Doric.  The  triglyph  is  placed  over  every  column,  and 
one  or  more  intermediately  over  every  intercolumn  (or  span  between  two 


ARCHITECTURAL  DRAWING. 


567 


568 


ARCHITECTURAL  DRAWING. 


ARCHITECTURAL  DRAWING.  569 

columns),  at  such  a  distance  from  eacli  other  that  the  metopes,  c,  or  spaces 
between  the  triglyphs,  are  square. 

In  the  best  Greek  examples  of  the  order,  there  is  only  a  single  triglyph  over 
each  intercolumn.  The  end  triglyphs  are  placed  quite  up  to  the  edge  or  outer 
angle  of  the  frieze.  The  mutules  are  thin  plates  attached  to  the  under  side  or 
soffit  of  the  corona,  over  each  triglyph  and  each  metope,  with  the  former  of 
which  they  correspond  in  breadth,  and  their  soffits  or  under  surfaces  are 
wrought  into  three  rows  of  guttae  or  drops,  conical  or  otherwise  shaped,  each 
row  consisting  of  six  guttae,  or  the  same  number  as  those  beneath  each  triglyph. 
The  shaft  of  the  Doric  column  was  generally  fluted  ;  the  number  of  channels 
is  either  sixteen  or  twenty,  afterward  increased  in  the  other  orders  to  twenty- 
four,  a  center  flute  on  each  side  of  the  column. 

Figs.  1291  to  1294  exhibit  an  example  of  the  Ionic  order,  taken  from  the 
Temple  of  Minerva  Polias,  at  Athens.  Fig.  1291  is  an  elevation  of  the  capital 
and  entablature ;  Fig.  1292,  of  the  base;  Fig.  1293  is  a  sectional  half  of  the 
plan  of  the  column  at  the  base  and  the  top  ;  Fig.  1294  an  elevation  of  the  bal- 
uster side  of  the  capital.  It  differs  from  the  Doric  in  the  more  slender  pro- 
portions of  its  shaft,  and  the  addition  of  a  base ;  but  the  capital  is  the  indi- 
cial  mark  of  the  order. 

When  a  colonnade  was  continued  in  front  and  along  the  flanks  of  the  build- 
ing, this  form  of  capital  in  the  end  column  occasioned  an  offensive  irregularity ; 
for  while  all  the  other  columns  on  the  flanks  showed  the  volutes,  the  end  one 
showed  the  baluster  side.  It  was  necessary  that  the  end  column  should,  there- 
fore, have  two  adjoining  volute  faces,  which  was  effected  by  placing  the  volute 
at  the  angle  diagonally. 

Figs.  1295  and  1296  represent  an  example  of  the  Corinthian  order,  from 
the  Arch  of  Hadrian,  at  Athens.  This  order  is  distinguished  from  the  Ionic 
more  by  its  deep  and  foliaged  capital  than  by  its  proportions.  The  capital  is 
considerably  more  than  a  diameter  in  height,  varying  in  different  examples 
from  one  to  one  and  a  half  diameter,  upon  the  average  about  a  diameter  and  a 
quarter,  and  has  two  rows  of  leaves,  eight  in  each  row,  so  disposed  that  of  the 
taller  ones,  composing  the  upper  row,  one  comes  in  the  middle,  beneath  each 
face  of  the  abacus,  and  the  lower  leaves  alternate  with  the  upper  ones,  coming 
between  the  stems  of  the  latter  ;  so  that  in  the  first  or  lower  tier  of  leaves  there 
is  in  the  middle  of  each  face  a  space  between  two  leaves  occupied  by  the  stem 
of  the  central  leaf  above  them.  Over  these  two  rows  is  a  third  series  of  eight 
leaves,  turned  so  as  to  support  the  small  volutes  which,  in  turn,  support  the 
angles  of  the  abacus.  Besides  these  outer  volutes,  invariably  turned  diagonally, 
there  are  two  other  smaller  ones,  termed  caulicoli,  which  meet  each  other  be- 
neath a  flower  on  the  face  of  the  abacus.  The  sides  of  the  abacus  are  concave 
in  plan,  being  curved  outward  so  as  to  produce  a  sharp  point  at  each  corner, 
which  is  usually  cut  off. 

Fig.  1297  represents  one  of  the  capitals  of  the  Tower  of  the  Winds,  showing 
the  earliest  formation  of  the  Corinthian  capital.  In  this  example  the  abacus 
is  square,  and  the  upper  row  of  leaves,  of  the  kind  called  water-leaves,  are 
broad  and  flat,  and  merely  carved  upon  the  vase  or  body  of  the  capital. 

The  shaft  is,  in  general,  fluted,  similarly  to  that  of  the  Ionic  column,  but 


570 


ARCHITECTURAL   DRAWING. 


ARCHITECTURAL  DRAWING. 


571 


FIG.  1297. 


sometimes  the  flutes  are  cabled;  that  is,  the  channels  are  hollowed  out  for  only 
about  two  thirds  of  the  upper  part  of  the  shaft,  and  the  remainder  cut  so  that 
each  channel  has  the  appearance  of  being  partly 
filled  up  by  a  round  staff  or  piece  of  rope,  v^  t  * 

The  cornice  is  very  much  larger  than  in  the 
other  orders,  in  height  and  in  projection,  consist- 
ing of  a  greater  number  of  moldings  beneath  the 
corona,  for  that  and  the  cymatium  over  it  are  in- 
variably the  crowning  members.  In  Fig.  1295 
square  blocks  or  dentels  are  introduced,  but  often 
to  the  dentels  is  added  a  row  of  modillions  (Fig. 
1418),  immediately  beneath,  and  supporting  the 
corona ;  and  between  them  and  the  dentels,  and 
also  below  the  latter,  are  other  moldings,  some- 
times cut,  at  others  left  plain. 

The  Composite  Order  is  a  union  of  the  Ionic  and  Corinthian  orders.  Its 
capital  consists  of  a  Eoman  Ionic  one,  superimposed  upon  a  Corinthian 
foliaged  base,  in  which  the  leaves  are  without  stalks,  placed  directly  upon  the 
body  of  the  vase. 

The  spacing  between  the  columns,  or  intercolumn,  is  from  one  to  one  and 
one  half  diameters,  but  modern  architects  have  coupled  the  columns,  making 
a  wide  intercolumn  between  every  pair  of  columns,  so  that  as  regards  the 
average  proportion  between  solids  and  voids,  that  disposition  does  not  differ 
from  what  it  would  be  were  the  columns  placed  singly.  Supercolumniation, 
or  the  system  of  piling  up  orders,  or  different  stages  of  columns  one  above  an- 
other, was  employed  for  such  structures  merely  as  were  upon  too  large  a  scale 
to  admit  of  the  application  of  columns  at  all  as  their  decoration,  otherwise  than 
by  disposing  them  in  tiers. 

The  Greeks  seldom  employed  human  figures  to  support  entablatures  or 
beams  ;  the  female  figures,  or  Caryatides,  are  almost  uniformly  represented 

in  an  erect  attitude,  without  any  apparent 
effort  to  sustain  any  load  ;  while  the  male  fig- 
ures, Telamones  or  Atlantes,  display  strength 
and  muscular  action.  Besides  entire  figures, 
either  Hermes'  pillars  or  Termini  are  occa- 
sionally used  as  substitutes  for  columns  of 
the  usual  form,  on  a  moderate  scale.  The 
first  mentioned  consist  of  a  square  shaft  with 
a  bust  or  human  head  for  its  capital ;  the  lat- 
ter of  a  half-length  figure  rising  out  of,  or  ter- 
minating in,  a  square  shaft  tapering  down- 
ward. Hermes'  pillars  are  frequently  em- 
ployed by  modern  architects  for  the  decora- 
tion of  window  architraves. 

The  Romans  introduced  circular  forms  and 

curves,  not  only  in  elevation  and  section,  but  in  plan.     The  true  Roman  order 
consists,  not  in  any  of  the  columnar  ordinances,  but  in  an  arrangement  of 


572 


ARCHITECTURAL  DRAWING. 


timOOBAIflBromiAW^ 


FIG.  1300. 


two  pillars  (Fig.  1298)  placed  at  a  distance  from  one  another  nearly  equal  to 
their  own  height,  and  having  a  very  long  entablature,  which,  in  consequence, 
required  to  be  supported  in  the  center  by  an  arch  springing  from  piers. 

Figs.  1299,  1300,  and 
1301,  from  the  Palace  of 
Diocletian  at  Spalatro,  are 
illustrations7 of  the  differ- 
ent modes  of  treatment  of 
the  arch  and  entablature. 

Perhaps  the  most  satis- 
factory works   of   the   Ro- 
mans  are   those  which  we 
consider  as  belonging  to  civil 
engineering  rather  than  to 
architecture  —  their    aque- 
ducts and  viaducts,  all  of 
which,  admirably  conceived  and 
executed,  have  furnished  practi- 
cal   examples   for   modern   con- 
structions,   of   which   the    High 
Bridge  across  Harlem  Eiver  may 
be  taken  as  an  illustration. 

The  history  of  Roman  archi- 
tecture is  that  of  a  style  in  course  of 
transition,  beginning  with  purely  pagan 
or  Grecian,  and  passing  into  a  style 
almost  wholly  Christian.  The  first 
form  of  Christian  art  was  ths  Roman- 
esque, which  afterward  branched  off 
into  the  Byzantine  and  the  Gothic. 

The  Romanesque  and  Byzantine,  as 
far  as  regards  the  architectural  features, 
are  almost  synonymous ;  in  the  earlier 
centuries  there  is  an  ornamental  dis- 
tinction. In  its  widest  signification, 
the  Romanesque  is  applied  to  all  the 

earlier  round-arch  developments,  in  contradistinction  to  the  Gothic  or  later 
pointed  arch  varieties  of  the  North.  In  this  view  the  Norman  is  included  in 
the  Romanesque. 

The  general  characteristics  of  the  Gothic  are  its  essentially  pointed  or  ver- 
tical tendency,  its  geometrical  details,  its  window- tracery,  its  openings,  its 
cluster  of  shafts  and  bases,  its  suits  of  moldings,  the  universal  absence  of  the 
dome,  and  the  substitution  of  the  pointed  for  the  round  arch. 

The  Romanesque  pillars  are  mostly  round  or  square,  and,  if  square,  gener- 
ally set  evenly,  while  the  Gothic  square  pillar  is  set  diagonally. 

Figs.  1302  to  1306  represent  sections  of  Gothic  pillars.  Fig.  1307  is  half 
of  one  of  the  great  western  piers  of  the  Cathedral  of  Bourges,  measuring  8  feet 


ARCHITECTURAL  DRAWING. 


573 


FIG.  1302. 


FIG.  1303. 


FIG.  1304. 


FIG.  1305. 


FIG.  1306. 


1 

\  — 

1 

r 

J  —  j 

i 

a 

i 

-v- 

=) 

1 

1 

B 

1 

FIG.  1307. 


FIG.  1308. 


FIG.  1309. 


FIG.  1311. 


FIG.  1313. 


FIG. 

1314. 

574: 


ARCHITECTURAL  DRAWING. 


on  each  side.  Figs.  1308  and  1309  are  elevations  of  capitals  and  bases,  and  sec- 
tions of  Gothic  pillars  ;  one  from  Salisbury,  the  other  from  Lincoln  Cathedral. 
Figs.  1310,  1311,  and  1312  are  examples  of  Byzantine  capitals  ;  Fig.  1313 
a  Norman  one,  from  Winchester  Cathedral  ;  and  Fig.  1314  a  Gothic  capital 
and  base,  from  Lincoln  Cathedral. 


FIG.  1315. 


FIG.  1316. 


FIG.  1317. 


Arches  are  generally  divided  into  the  triangular-headed  arch,  the  round- 
headed  arch,  and  the  pointed  arch.     Of  the   round-headed  arch,   there  are 
semicircular,   segmental,  stilted  (Fig.  1315),  and  horseshoe  (Fig.  1316).     Of 
the  two-centered  pointed,  the  equilateral  (Fig.  1317),  the  lancet,  and  the  ob- 
tuse.    Of  the  first,  the  radii  of  the  seg- 
ments forming  the  arch  are  equal  to  the 
breadth  of  the  arch,  of  those  of  the  lan- 
cet longer,  and  of  the  obtuse  shorter. 

Of  the  complex  arches,  there  are  the 
ogee  (Fig.  1318)  and  the  Tudor  (Fig. 
1319).  The  Tudor  arch  is  described  from 
four  centers,  two  on  a  level  with  the 
spring  and  two  below  it. 

Of  foiled  arches,  there  are  the  round-headed  trefoil  (Fig.  1320),  the  pointed 
trefoil  (Fig.  1321),  and  the  square-headed  trefoil  arch  (Fig.  1322).  The  points 
c  are  termed  cusps. 

The  semicircular  arch  is  the  Koman  Byzantine  and  Norman  arch ;  the  ogee 


FIG.  1318. 


FIG.  1319. 


FIG.  1320. 


FIG.  1321. 


FIG.  1322. 


and  horseshoe  are  the  profiles  of  many  Turkish  and  Moorish  domes;  the  pointed 
and  foliated  arches  are  Gothic. 

Domes  and  Vaults. — The  Greek  vaulting  consisted  wholly  of  spherical  sur- 
faces, the  Koman  of  cylindrical  ones.  Figs.  1323  and  1324  illustrate  this  dis- 
tinction, Fig.  1323  being  the  elevation  of  a  Roman  cylindrical  cross-vault,  and 
Fig.  1324  the  elevation  of  the  roof  of  the  church  of  St.  Sophia  at  Constantino- 
ple ;  and  the  sprouting  of  domes  out  of  domes  continues  to  characterize  the 
Byzantine  style.  As  a  constructive  expedient  the  cross-vault  is  to  be  preferred, 


ARCHITECTURA1 

as  the  whole  pressure  and  thrust  are  colL 
plied  at  the  angles  only,  so  that  it  might  be 

placed  in 
strong  enough 


AWING. 


•: 


no 


in  four  definite  resultants,  a; 
rted  by  four  flying  buttresses, 
3tioii  of  these  resultants,  n 
rushed  by  the  p: 


FIG.  1323. 


FIG.  1324. 


FIG.  1325. 


Fig.  1325  represents  a  compartment  of  the  simplest  Gothic  vaulting — a,  a, 
groin  ribs  ;  #,  I,  b,  side  ribs. 

The  Romans  introduced  side  ribs,  appearing  on 
the  inside  as  flat  bands,  and  harmonizing  with  the 
similar  form  of  pilasters  in  the  walls,  but  they  never 
used  groin  ribs ;  the  Gothic  builders  introduced 
these,  and  deepened  the  Roman  ribs.  The  impene- 
tration  of  vaults,  either  round  or  pointed,  produces 
elliptical  groin  lines,  or  else  lines  of  double  curva- 
ture ;  yet  the  early  Gothic  architects  made  their 
groin  ribs  usually  simple  pointed  arches  of  circular 
curvature,  thrown  diagonally  across  the  space  to  be 

groined,  and  the  four  side  arches  were  equally  simple,  the  only  care  being  that 
all  the  arches  should  have  their  vertices  at  the  same  level.  The  strength  de- 
pended on  the  ribs,  and  the  shell  was  made  quite  light,  often  not  more  than 
six  inches,  while  Roman  vaults  of  the  same  span  would  have  been  three  or  four 
feet.  The  Romans  made  their  vault  surfaces  geometrically  regular,  and  left 
the  groins  to  take  their  chance  ;  while  the  early  Gothic  architects  made  their 
groins  geometrically  regular,  and  let  the  intermediate  surfaces  take  their  chance. 

In  the  next  step  the  groin  ribs  were  elliptical,  and  when  intermediate  ribs 
or  tiercerons  were  inserted,  these  ribs  had  also  elliptical  or  cylindrical  curva- 
tures, diiferent  from  the  groins,  and  the  ribs  were  placed  near  each  other,  in 
order  that  the  portion  of  the  vault  between  each  pair  might  practically  be 
almost  cylindrical.  In  the  formation  of  the  compound  circular  ribs  three  con- 
ditions were  to  be  observed  :  1.  That  the  two  arcs  should  have  a  common  tan- 
gent at  the  point  of  meeting.  2.  That  the  feet  of  all  the  ribs  should  have  the 
same  radius,  up  to  the  level  at  which  they  completely  separate  from  each  other. 
3.  That  from  this  point  upward  their  curvature  should  be  so  adjusted  as  to 
make  them  all  meet  their  fellows  on  the  same  horizontal  plane,  so  that  all  the 
ridges  of  the  vaults  may  be  on  one  level. 

The  geometrical  difficulty  of  such  works  led  to  what  is  called  fan-tracery 
vaulting.  If  similar  arches  spring  from  each  side  of  the  pillars  (Fig.  1325), 
the  portion  of  vault  springing  from  each  pillar  would  have  the  form  of  an  in- 
verted concave-sided  pyramid,  its  horizontal  section  at  every  level  being  square. 
The  later  architects,  by  converting  this  section  into  a  circle,  the  four-sided 


576 


ARCHITECTURAL   DRAWING. 


FIG.  1326. 


pyramid  became  a  conoid,  and  all  the  ribs  forming  the  conoidal  surface  became 
alike  in  curvature,  so  that  they  all  might  be  made  simple  circular  arcs  ;  these 
ribs  are  continued  with  unaltered  curvature  till  they  meet  and  form  the  ridge  ; 

but  in  this  case  the  ridges  are  not  level,  but 
gradually  descend  every  way  from  the  center 
point  (Fig.  1326). 

In  the  figure  this  is  not  fully  carried  out, 
for  no  rib  is  continued  higher  than  those 
over  the  longer  sides  of  the  compartment,  so 
that  a  small  lozenge  is  still  left,  with  a  boss 
at  its  center.  When  the  span  of  the  main 
arch  1)  a  was  large  in  proportion  to  that  of 
b  c,  the  arch  b  c  became  a  very  acute  lancet 
arch,  scarcely  admitting  windows  of  an  ele- 
gant or  sufficient  size.  To  obviate  this,  the 
compound  curve  was  again  introduced. 

The  four-centered  arch  is  not  necessarily  flat  or  depressed  ;  it  can  be  made 
of  any  proportion,  high  or  low,  and  always  with  a  decided  angle  at  the  vertex. 
In  general,  the  angular  extent  of  the  lower  curve  is  not  more  than  65°,  nor  less 
than  45°.  The  radius  of  the  upper  curve  varies  from  twice  to  more  than  six 
times  the  radius  of  the  lower.  The  projecting  points  of  the  trefoil  arch,  or 
cusps,  are  often  introduced  for  ornament  merely,  but  serve  constructively,  both 
in  vaults  and  arches,  as  a  load  for  the  sides,  to  prevent  them  rising  from  the 
pressure  on  the  crown. 

As  vaultings,  in  general,  were  contrived  to  collect 
the  whole  pressure  of  each  compartment  into  four  sin- 
gle resultants,  at  the  points  of  springing,  leaving  the 
walls  so  completely  unloaded  that  they  are  required 
only  as  inclosures  or  screens,  they  might  be  entirely 
omitted  or  replaced  by  windows.  Indeed,  the  real  sup- 
porting walls  are  broken  into  narrow  strips,  placed  at 
right  angles  to  the  outline  of  the  building,  and  called 
buttresses,  and  the  inclosing  walls  may  be  placed  either 
at  the  outer  or  inner  edge  of  the  buttresses.  The  first, 
that  adopted  by  the  French  architects,  gave  deep  re- 
cesses to  the  interiors,  while  the  other,  or  English 
method,  served  to  produce  external  play  of  light  and 
shade. 

The  Norman  buttress  (Fig.  1327)  resembles  a  flat 
pilaster,  being  a  mass  of  masonry  with  a  broad  face, 
slightly  projecting  from  the  wall.  They  are,  generally, 
of  but  one  stage,  rising  no  higher  than  the  cornice, 
under  which  they  often,  but  not  always,  finish  with  a 
slope.  Sometimes  they  are  carried  up  to,  and  terminate 
in,  the  corbel  table. 

Fig.  1328  represents  a  buttress  in  two  stages,  with  slopes  as  set-offs. 

Fig.  1329  is  a  buttress  of  the  Early  English  style,  having  a  plain  triangular 


FIG.  1327. 


ARCHITECTURAL  DRAWING. 


577 


FIG.  1333. 


or  pedimental  head.  The  angles  were  sometimes  chamfered 
off,  and  sometimes  ornamented  with  slender  shafts.  In  but- 
tresses of  different  stages,  the  triangular  head  or  gable  is  used 
as  a  finish  for  the  intermediate  stages. 

In  the  Decorated  style,  the  outer  surfaces  of  the  buttresses 
are  ornamented  with  niches,  as  in  Fig.  1330.  In  the  Perpen- 
dicular style  the  outer  surface  is  often  partially  or  wholly  cov- 
ered with  panel-work  tracery  (Fig.  1331). 

The  buttress  was  a  constructive  expedient  to  resist  the 
thrust  of  vaulting ;  but  to  resist  the  thrust  of  the  principal 
vault,  or  that  over  the  nave  or  central  part  of  the  church, 
buttresses  of  the  requisite  depth  would  have  filled  up  the  side 


FIG.  1329. 


FIG.  1330. 


FIG.  1331. 


FIG.  1332. 


aisles  entirely.  To  obviate  this,  the  system  of  flying  but- 
tresses was  adopted ;  that  is,  the  connection  of  the  interior 
with  the  outer  buttress,  by  an  arch  or  system  of  arches,  as 
shown  in  Fig.  1332.  The  outer  piers  were  surmounted  by 
pinnacles,  to  render  them  a  sufficiently  steady  abutment  to 
the  flying  arches. 

The  earlier  towers  of  the  Romanesque  style  were  construct- 
ed without  spires.  All  are  square  in  plan,  and  extremely 
similar  in  design.  Fig.  1333  is  an  elevation  of  the  tower 
attached  to  the  church  of  Sta.  Maria,  in  Cosmedin,  and  is  one 
of  the  best  and  most  complete  examples  of  this  style.  It  is 
15  feet  broad  and  110  feet  high.  These  towers  are  the  types 
of  the  later  Italian  campaniles,  generally  attached  to  some 


578 


ARCHITECTURAL  DRAWING. 


angle  of  churches ;  if  detached,  so  placed  that 
they  still  form  a  part  of  the  church  design. 
Sometimes  they  are  but  civic  constructions,  as 
belfries,  or  towers  of  defense.  The  campanile  is 
square,  carried  up  without  break  or  offset  to  two 
thirds,  at  least,  of  its  intended  height ;  it  is  gen- 
erally solid  to  a  considerable  height,  or  with  only 
such  openings  as  serve  to  admit  light  to  the  stair- 
cases. Above  this  solid  part  one  round  window 
is  introduced  in  each  face  ;  in  the  next  story, 
two  ;  in  the  one  above  this,  three ;  then  four, 
and  lastly  five ;  the  lights  being  separated  by 
slight  piers,  so  that  the  upper  story  is  virtually 
an  open  loggia. 

The  Gothic  towers  have  projecting  buttress- 
es, frequent  offsets,  lofty  spires,  and  a  general 


FIG.  1334. 


FIG.  1335. 

pyramidal  form.  Fig.  1334  is  the  front  eleva- 
tion of  a  simple  English  Gothic  tower ;  here  the 
plain  pyramidal  roof,  rising  at  an  equal  slope  on 
eacji  of  the  four  sides,  is  intersected  by  an  octag- 
onal spire  of  steep  pitch.  The  first  spires  were 
simple  quadrangular  pyramids  ;  afterward  the  an- 
gles were  cut  off,  and  they  became  octagonal,  and 
this  is  the  general  Gothic  form  of  spire.  Often, 
instead  of  intersecting  the  square  roof,  as  in  the 
figure,  the  octagonal  spire  rests  upon  a  square 
base,  and  the  angles  of  the  tower  are  carried  up 
by  pinnacles,  or  the  sides  by  battlements,  or  by 
both,  as  in  Fig.  1335,  to  soften  the  transition  be- 
tween the  perpendicular  and  sloping  part. 

In  general  the  spires  of  English  churches  are 
more  lofty  than  those  on  the  Continent ;  the 
angle  at  the  apex  in  the  former  being  about  10°, 
and  in  the  latter  about  15°.  The  apex  angle  of 


ARCHITECTURAL  DRAWING. 

Fm.  1338.         FIG.  1339.   FIG.  1337- 


579 


FTG.  1341, 


FIG.  1342.       FIG.  1343. 


FIG.  1340. 


FIG.  1344. 


580 


ARCHITECTURAL  DRAWING. 


the  spires  of  Chichester  and  Lichfield  are  from  12°  to  13°,  or  a  mean  between 
the  two  proportions,  and,  according  to  Ferguson,  more  pleasing  than  either. 
Although  having  more  lofty  spires,  yet  the  English  construction  is  much  more 
massive  in  appearance  than  the  Continental ;  the  apertures  are  less  numerous, 

and  the  surfaces  are  less  cut 
up  and  covered  with  orna- 
ments. The  spires  of  Fri- 
berg  Church,  and  many  oth- 
ers on  the  Continent,  are 
open  work. 

Figs.  1336  and  1337  are 
bell-cots.  Figs.  1338  to  1344 
are  spires.  Fig.  1345  is  an 


FIG.  1 


FIG.  1347. 

is  applied 
service  of 


FIG.  1349. 

apse,  or  circular  end  of  a 
church,  from  German  Gothic 
examples. 

Figs.  1346  and  1347  are 
examples  of  spire  finials,  with 
weather-cocks. 

Figs.  1348  and  1349  are 
examples  of  towers  not  con- 
nected with  church  edifices. 

Fig.  1350  is  a  tower  of 
very  recent  construction,  and 

to  the  utilitarian  purpose  of  sustaining  a  water-tank  for  the  highest 
the  Croton  in  New  York  city. 


FIG.  1348. 


ARCHITECTURAL  DRAWING. 


581 


FIG.  1350. 


Fig.  1351  represents  the  upper  portion  of  the  tower  of 
Ivan  Veliki,  at  Moscow.  The  Russian  towers  are  generally 
constructed  independent  of  their  churches,  and  are  intend- 
ed for  the  reception  of  their  massive  bells. 


FIG.  1352. 


Pllnllil 


FIG.  1351. 


FIG.  1353. 


Windows. — Before  the  use  of  painted  glass,  as  very 
small  apertures  sufficed  for  the  introduction  of  the  required 
quantity  of  light  into  a  church,  the  windows  of  the  Roman- 
esque  churches  were  generally  small,  and  devoid  of  tracery  ; 
and  as  the  Byzantine  architects,  adorning  the  walls  with 
paintings,  could  not  use  stained  glass,  they  followed  in  gen- 
eral form  the  Romanesque  window,  apertures  with  circular  heads,  either  single 
or  in  groups  (Fig.  1353  or  Fig.  1352).  The  Norman  windows  were  also  small, 
each  consisting*  of  a  single  light,  semicircular  in  the  head,  and  placed  as  high 
as  possible  above  the  ground  ;  at  first  splayed  on  the  inside  only,  afterward 
the  windows  began  to  be  recessed  with  moldings  and  jamb-shafts  in  the  angles, 
as  in  Fig.  1353. 

The  Lancet,  in  general  use  in  the  early  Gothic  period,  was  of  the  simplest 
arrangement :  in  these  windows  the  glass  was  brought  within  three  or  four 
inches  of  the  outside  of  the  wall,  and  the  openings  were  widely  splayed  in  the 
interior.  The  proportions  of  these  windows  vary  considerably  ;  in  some  the 
height  being  but  five  times  the  width,  in  others  as  much  as  eleven ;  eight  or 
nine  times  may  be  taken  as  the  average.  Lancet  windows  occur  singly  (Fig. 
1354),  or  in  groups  of  two,  three,  five,  and  seven,  rarely  of  four  and  six.  The 
triplet  (Fig.  1355)  is  the  most  beautiful  arrangement  of  lancet  windows.  It 


582 


ARCHITECTUEAL  DRAWING. 


was  customary  to  mark  with  greater  importance  the  central  light,  by  giving  it 
additional  height,  and  in  most  cases  increased  width  also.  In  some  examples 
the  windows  of  a  lancet  triplet  are  placed  within  one  drip-stone  forming  a  sin- 


FIG.  1355. 


FIG.  1354. 


FIG.  1356. 


gle  arch,  thus  bearing  a  strong  resemblance  to  a  single  three-light  window. 
The  first  approximation  to  tracery  appears  to  have  been  the  piercing  of  the  space 
over  a  double  lancet  window  comprised  within  a  single  drip-stone  (Fig.  1356). 

A  traceried  window  is  a  distinctive  characteristic  of  Gothic  architecture ; 
with  the  establishment  of  the  principle  of  window  tracery  the  mullions  were 
recessed  from  the  face  of  the  wall  in  which  the  window  arch  was  pierced,  and 
the  fine  effect  thus  produced  was  speedily  enhanced  by  the  introduction  of  dis- 
tinct orders  of  mullions,  and  by  recessing  certain  portions  of  the  tracery  from 
the  face  of  the  primary  mullions  and  their  corresponding  tracery  bars. 

Decorated  window  tracery  is  divided  into  two  chief  varie- 


FIG.  1357. 


FIG.  1358. 


FIG.  1359. 


ties,  Geometrical  and  Flowing ;  the  former  consisting  of  geometrical  figures, 
as  circles,  trefoils,  quatrefoils,  curvilinear  triangles,  lozenges,  etc.  ;  while  in 
flowing  tracery  these  figures,  though  still  existing,  are  gracefully  blended  to- 
gether in  one  design. 


ARCHITECTURAL   DRAWING. 


583 


Fig.  1357  represents  a  quatrefoil  window,  Fig.  1358  a  pointed  trefoil  in  out- 
line with  the  centers  of  the  different  circles  and  such  constructive  lines  indi- 
cated as  may  be  necessary.  Fig.  1359  represents  two  forms  of  circular  win- 
dows, or  roses  tournantes. 

Fig.  1360  represents  an  example  of  the  earlier  decorated  tracery  window- 
head,  consisting  of  two  foiled  lancets,  with  a  pointed  quatrefoil  in  the  span- 
drel  between  them.     One  half  of 
the  windows  in  this,  as  in  some  of 


FIG.  1360. 


FIG.  1361. 


the  following  figures,  is  drawn  in   skeleton,  to   explain   their  construction. 
Fig.  1361  is  another  example  of  Decorated  tracery. 

Fig.  1362  is  an  example  of  the  English  leaf  tracery ;  Fig.  1363,  of  the 
French  flamboyant.     The  difference  between  the  two  styles  is,  that  while  the 
upper  ends  of  the  English  loops  or  leaves  are 
round,  or  simply  pointed ;  the  upper  ends  of 


FIG.  1362. 


FIG.  1363. 


FIG.  1364. 


the  latter  terminate,  like  their  lower  ones,  in  angles  of  contact,  giving  a  flame- 
like  form  to  the  tracery  bars  and  form  pieces. 

In  England  the  Perpendicular  style  succeeded  the  Decorated  ;  the  mullions, 
instead  of  diverging  in  flowing  or  curvilinear  lines,  are  carried  up  straight 
through  the  head  of  the  windows ;  smaller  mullions  spring  from  the  head  of 
the  principal  lights,  and  thus  the  upper  portion  of  the  window  is  filled  with 
panel-like  compartments.  The  principal  as  well  as  the  subordinate  lights  are 
foliated  in  their  heads,  and  large  windows  are  often  divided  horizontally  by 
transoms.  The  forms  of  the  window  arches  vary  from  simple  pointed  to  the 
complex  four-centered,  more  or  less  depressed. 

Fig.  1364  is  an  example  of  a  Perpendicular  window. 


584 


ARCHITECTURAL  DRAWING. 


FIG.  1365. 


FIG.  1366. 


FIG.  1367. 


FIG.  1368. 


Fig.  1365  is  a  square-headed  window,  such 
as  were  usual  in  the  clear-stories  of  Perpen- 
dicular architecture. 

Figs.  1366  and  1367  are  quadrants  of  cir- 
cular windows,  used  more  especially  in  France, 
for  the  adornment  of  the  west  ends  and  tran- 
septs of  the  cathedrals. 

Besides  the  tracery  characteristic  of  Gothic 
architecture,  there  is  a  tracery  peculiar  to  the 
Saracenic  and  Moorish  style,  of  which  Fig. 
1368  may  be  taken  as  an  example — it  being  a 
window  of  one  of  the  earliest  mosques.  The 
general  form  of  the  window  and  door-heads  of 


this  style  is  that  of  the  horse-shoe,  either  circular  or  pointed. 

Doorways. — Fig.   1369  is  the  eleva- 
tion of  a  circular-headed  doorway,  which 


FIG.  1370. 


FIG.  1369. 


FIG.  1371. 


ARCHITECTURAL  DRAWING. 


585 


may  be  considered  the  type  of  many  entrances  both  in  Romanesque,  Gothic, 
and  later  styles.  It  consists  of  two  or  more  recessed  arches,  with  shafts  or 
moldings  in  the  jambs.  In  the  earlier  styles  the  arches  were  circular,  in  the 
later  Gothic,  generally  pointed,  but  sometimes  circular  ;  in  the  earlier,  the 
angles  in  which  the  shafts  are  placed  are  rectangular ;  in  the  later,  the  shaft 
is  often  molded  on  a  chamfer  plane,  that  is,  a  plane  inclined  to  the  face  of  the 
wall,  generally  at  an  angle  of  45°  ;  often  the  chamfer  and  rectangular  planes 
are  used  in  connection. 

Fig.  1370  is  a  simple  head  of  a  depressed  four-centered  or  Tudor-arched 
doorway,  with  a  hood  molding. 

Fig.  1371  represents  the  incorporation  of  a  window  and  doorway.  Some- 
times the  doorway  pierces  a  buttress ;  in  that  case,  the  buttress  expands  on 
either  side,  forming  a  sort  of  porch.  The  Gothic  architects  placed  doors 
where  they  were  necessary,  and  made  them  subservient  to  the  beauty  of  the 
design. 

Fig.  1372  is  an  example  of  a  gabled  doorway  with  crockets  and  finial. 


FIG.  1372. 


FIG.  1373. 


Fig.  1373  is  an  example  of  a  perpendicular  doorway,  with  a  label  or  hood 
molding  above,  and  ornamented  spandrels. 

Fig.  1374  is  an  example  of  a  Byzantine,  and  Fig.  1375  of  a  Saracenic 
doorway. 

The  Renaissance  style  was,  originally,  but  the  revival  or  a  fair  rendering  of 
the  classical  orders  of  architecture,  with  ornaments  from  the  Byzantine  and 
Saracenic  styles. 

Garbett  divides  this  style  into  three  Italian  schools,  the  Florentine,  Vene- 
tian, and  Roman.  The  Florentine  admits  of  little  apparent  ornament,  but 
any  degree  of  real  richness,  preserving  in  its  principal  forms  severe  contrast ; 
powerful  masses  self-poised  without  corbeling,  without  arching  ;  breadth  of 
everything,  of  light,  of  shade,  of  ornament,  of  plain  wall ;  depth  of  recess  in 
the  openings,  of  perspective  in  the  whole  mass,  of  projection  in  the  cornice. 
Absence  of  features  useless  to  convenience  or  stability,  admitting  of  great 
plainness,  or  of  very  florid  enrichment. 


586 


ARCHITECTURAL  DRAWING. 


The  aim  of  the  Venetian  school  was  splendor,  variety,  show,  and  ornament ; 
not  so  much  real  as  effective  ornament.  Thus,  it  rarely  contains  as  much 
carving  or  minute  enrichment  as  the  Florentine  admits ;  but  it  has  larger 
ornaments,  constructed  (or  built)  ornaments,  great  features  useless  except  for 
ornament,  such  as  inaccessible  porticoes,  detached  columns,  and  architraves 
supporting  no  ceiling,  towers  built  only  for  breaking  an  outline. 


FIG.  1374. 


FIG.  1375. 


The  Roman  school  is  intermediate  in  every  respect  between  the  two  other 
schools.  It  is  better  adapted  to  churches  than  to  any  other  class  of  buildings. 
This  fitness  arises  from  the  grand,  simple,  and  unitary  effect  of  one  tall  order, 
generally  commencing  at  or  near  the  ground,  obliterating  the  distinction  of 
two  or  three  stories,  making  a  high  building  appear  a  single  story. 

Moldings. — "All  classical  and  Romanesque  architecture  is  composed  of 
bold  independent  shafts,  plain  or  fluted,  with  bold  detached  capitals  forming 
arcades  or  colonnades  where  they  are  needed,  and  of  walls  whose  apertures  are 
surrounded  by  courses  of  parallel  lines  called  moldings, 
and  have  neither  shafts  nor  capitals.  The  shaft  system 
and  molding  system  are  entirely  separate  ;  the  Gothic 
architects  confounded  the  two  ;  they  clustered  the  shafts 
till  they  looked  like  a  group  of  moldings,  they  shod  and 
capitaled  the  moldings  till  they  looked  like  a  group  of 
shafts."  The  moldings  appear  in  almost  every  conceivable 
position  ;  from  the  bases  of  piers  and  piers  themselves,  to 
the  ribs  of  the  fretted  vaults  which  they  sustain. 

In  the  earliest  examples  of  Norman  doorways  the  jambs 
are  mostly  simply  squared  back  from  the  walls  ;  recessed 
jambs  succeeded,  and  are  common  in  both  Norman  and  Gothic  architecture  ; 
and  when  thus  recessed,  detached  shafts  were  placed  in  each  angle  (Fig.  1376). 


FIG.  1376. 


ARCHITECTURAL  DRAWING. 


587 


In  the  later  styles  the  shafts  were  almost  invariably  attached  to  the  structure. 
The  angles  themselves  were  often  cut  or  chamfered  off,  and  the  moldings 
attached  to  the  chamfer-plane.  The  arrangement  of  window  jambs,  during 
the  successive  periods,  was  in  close  accordance  with  that  of  doorways. 

In  the  richer  examples  small  shafts  were  introduced,  which,  rising  up  to 
the  springing  of  the  window,  carried  one  or  several  of  the  arch  moldings.     Yet 


FIG.  1385. 


FIG.  1384. 

moldings  are  nevertheless  not  essential  accessories ; 
many  windows  of  the  richest  tracery  have  their  mull- 
ions  and  jambs  composed  of  simple  chamfers. 

Figs.    1377   to    1385  are   examples   of  arch  and 

architrave  moldings,  which,  even  when  not  continuous,  partook  of  the  same 

general  arrangement  as  those  in  the  jambs,  with  greater  richness  of  detail. 

When  shafts  were  employed,  they  carried  groups  of  moldings  more  elaborate. 

than  those  of  the  jambs,  though  still  falling  on  the  same  planes. 


£88 


ARCHITECTURAL  DRAWING. 


Capitals  were  either  molded  or  carved  with  foliage,  animals,  etc.  ;  they 
always  consisted  of  three  distinct  parts  (Fig.  1386) — the  head  mold  (A),  the 
bell  (B),  and  the  neck  mold  (C).     In  Norman  examples  the 
head  mold  was  almost  invariably  square  ;  in  the  later  styles 
it  is  circular,  or  corresponding  to  the  form  of  the  pillar. 

Bases  consist  of  the  plinth  and  the  base  moldings.  The 
plinth  was  square  in  the  Norman  style,  afterward  octagonal ; 
then,  assuming  the  form  of  the  base  moldings,  it  bent  in 
and  out  with  the  outline  of  the  pier.  Base  moldings  were 
also  extensively  used  round  the  buttresses,  towers,  and  walls 
of  churches. 

String  Courses,  of  which  Figs.  1387  to  1392  are  exam- 
ples, were  horizontal  courses  in  the  face  of  a  wall ;  the  most 
usual  position  being  under  the  windows.  In  the  Norman  styles  they  were  usu- 
ally heavy  in  the  outline  ;  in  the  later  styles  they  were  remarkably  light  and 
elegant ;  free  from  restraint  or  horizontally  they  now  rose  close  under  the  sill 
of  the  window,  and  then  suddenly  dropping  to  accommodate  themselves  to  the 


FIG.  1386. 


FIG.  1387. 


FIG.  1388. 


FIG.  1389. 


FIG.  1390. 


FIG.  1391. 


FIG.  1392. 


arch  of  a  low  doorway,  and  again  rising  to  run  immediately  under  the  adjoin- 
ing window.  In  this  way  the  string  courses  frequently  served  the  purpose  of 
a  drip-stone  or  hood  molding  over  doors  ;  occasionally  the  hood  mold  was  con- 
tinued from  one  window  to  the  other. 

Cornices  are  not  an  essential  feature  in  Gothic  architecture.     In  the  Nor- 
man and  early  English  styles,  the  cornice  was  a  sort  of  enlarged,  projecting 
string  course,  forming  a  drip-stone  beneath  the  roof,  which,  if 
supported  on  brackets  or  corbels,  was  termed  the  corbel  table. 

The  earliest  molding  in  Norman  work  is  a  circular  bead 
strip,  worked  out  of  the  edges  of  a  recessed  arch,  called  a  cir- 
cular bowtel  (Fig.  1393).  From  a  circular  form  the  bowtel  soon 
became  pointed,  and,  by  an  easy  transition,  into  the  bowtel  of 
one,  two,  or  three  fillets. 

Figs.  1394  to  1399  are  sections  of  Eomanesque  drip-  or  cap- 
stones, adapted  to  different  pitches  of  roof. 


FIG.  1393. 


ARCHITECTURAL  DRAWING. 


FIG.  1397. 


FIG.  1398. 


Fig.  1400  is  the  scroll  molding ;  a  simple  filleted  bowtel,  with  the  fillet 
undeveloped  on  one  side,  as  shown  by  the  dotted  lines.  If  this  molding  be 
cut  in  half,  through  the  center  of  the  fillet,  we  have  on  the  developed  side  the 
molding  now  termed  by  carpenters  the  rule  joint,  which,  by  rounding  off  the 
corners  by  reverse  curves,  becomes  the 
wave  molding. 


FIG.  1400. 


Fig.  1401  is  a  Gothic  example  of 
the  filleted  bowtel  with  prominent 
alternate  hollows.  FIG.  1401.  FIG.  1402. 

Fig.  1402  is  an  example  of  the 
perpendicular  style,  an  insignificant  hollow  separating  groups  of  moldings. 


•590 


ARCHITECTURAL  DRAWING. 


Figs.  1403  to  1408  are  examples  of  molded  timbers,  used  largely  in  open- 
timbered  roofs  and  for  exposed  beams.  It  is  still  the  custom,  when  the  fram- 
ing is  not  covered  in  with  plastering  or  ceiling,  to  corner  the  edges  of  the  joists 
•and  beams,  at  an  angle  of  45°,  for  about  I"  on  each  face,  but  not  extending  it 
olose  to  the  joint  or  wall ;  this  is  called  stop-chamfering. 


FIG.  1403. 


FIG.  1405. 


FIG.  1407. 


FIG.  1408. 

Ornament. — Architectural  ornament  is  of  two  kinds,  constructive  and  deco- 
rative. By  the  former  is  meant  all  those  contrivances,  such  as  capitals,  brack- 
ets, vaulting-shafts,  and  the  like,  which  serve  to  explain  or  give  expression  to 
the  construction  ;  by  the  latter,  such  as  moldings,  frets,  foliage,  etc.,  which 
give  grace  and  life,  either  to  the  actual  constructive  form,  or  to  the  construct- 
ive decoration.  Moldings  of  the  different  styles  have  been  already  treated  of  ; 
it  is  proposed  to  give  now  what  are  even  more  purely  decorations  of  a  style. 

In  the  Grecian  orders  the  Doric  (Fig.  1286)  has  the  triglyph  mutules  and 
guttae  ;  the  Ionic  (Fig.  1291)  has  various  moldings  of  the  cornice,  frieze,  abacus, 
and  neck  of  the  column  enriched.  The  principal  ornament  of  the  neck  of  the 
column  is  the  anthemion,  commonly  known,  in  its  most  simple  form,  as  the 


FIG.  1409. 


FIG.  1410. 


honeysuckle  or  palmetto ;  in  the  anthemion,  as  represented  in  the  figure,  the 
palmetto  alternates  with  the  lily  or  some  analogous  form.  The  ornament 
of  the  abacus  is  the  egg  and  dart  (Fig.  1409)  ;  the  ornament  of  the  frieze  and 


ARCHITECTURAL  DRAWING. 


591 


'cornice  (Fig.  1410).     The  fret  (Fig.  1411)  and  the  guilloche  (Fig.   1412)  are 
also  common  Greek  ornaments,  used  to  adorn  the  soffits  of  beams  and  ceilings. 


FIG.  1413. 


FIG.  1414. 


The  acanthus  is  the  distinctive  ornament  of  the  Corinthian,  of  which  a  leaf  is 
represented  in  front  and  side  view  (Figs.  1413  and  1414). 


1415. 


FIG.  1416. 


FIG 


592 


ARCHITECTURAL  DRAWING. 


Figs.  1415,  1416,  and  1417  are  the  side  elevation,  front  elevation,  and  sec- 
tion of  a  Greek  bracket,  the  principal  ornaments  of  which  are  taken  from  the- 

anthemion  and  acanthus. 

Fig.  1418  is  an  elevation  of  a  por- 
tion of  an  enriched  cornice  from  the 


FIG.  1418. 


FIG.  1419. 


temple  of  Jupiter  Stator,  at  Rome,  of  the  Corinthian  order  of  architecture. 
Fig.  1419  is  the  under  side  of  the  modillion,  on  a  larger  scale. 

The  chief  characteristic  of  Roman  ornament  is  its  uniform  magnificence,  an 
enrichment  of  the  Greek.  The  most  used  elements  of  the  Roman  decorations, 
are  the  scroll  and  the  acanthus.  The  acanthus  of  the  Greeks  is  the  narrow 
prickly  acanthus ;  that  of  the  Roman,  the  soft  acanthus.  For  capitals  the 
Roman  acanthus  is  commonly  composed  of  conventional  clusters  of  olive-leaves. 
Fig.  1420  represents  a  Roman  acanthus  scroll. 


FIG.  1420. 

The  free  introduction  of  monsters  and  animals  is  likewise  a  characteristic 
of  Greek  and  Roman  ornament,  as  the  sphinx,  the  triton,  the  griffin,  and  oth- 
ers ;  they  occur,  however,  more  abundantly  in  the  Roman. 

Symbols  are  the  foundation  of  decorations  in  the  Byzantine  and  Romanesque. 
The  early  symbols  were  the  monogram  of  Christ,  the  lily,  the  cross,  the  ser- 
pent, the  fish,  the  aureole,  or  vesica  piscis,  and  the  circle  or  nimbus,  the  trefoil 
and  quatrefoil,  the  first  having  reference  to  the  Trinity,  the  second  to  the  four 
Evangelists.  Occasionally  the  symbolic  images  of  the  Evangelists,  the 


ARCHITECTURAL  DRAWING. 


593 


the  lion,  the  ox,  and  the  eagle,  are  represented  within  these  circles.  The  hand 
in  the  attitude  of  benediction,  and  the  lily  (the  fleur-de-lis),  the  emblem  of  the 
virgin  and  purity,  are  common  ;  also  a  peculiarly  formed  leaf,  somewhat  resem- 
bling the  leaf  of  the  ordinary  thistle.  The  serpent  figures  largely  in  Byzantine 
art  as  the  instrument  of  the  fall,  and  one  type  of  the  redemption. 

Pagan  ornaments,  under  certain  symbolic  modifications,  were  admitted  into 
Christian  decorations.  Thus  the  foliations  of  the  scroll  were  terminated  by 
lilies,  or  by  leaves  of  three,  four,  and  five  blades,  the  number  of  blades  being 
significant ;  and  in  a  similar  way  the  anthemion  and  every  other  ancient  orna- 
ment. In  the  Byzantine,  all  their  imitations  of  natural  forms  were  invariably 
conventional ;  it  is  the  same  even  with  animals  and  the  human  figure ;  every 
saint  had  his,  prescribed  colors,  proportions,  and  symbols. 


FIG.  1421. 


FIG.  1422. 

The  Saracenic  was  the  period  of  gorgeous  diapers  (Figs.  1421  and  1422),  for 
their  habit  of  decorating  the  entire  surfaces  of  their  apartments  was  highly  favor- 
able to  the  development  of  this  class  of  design.  The  Alhambra  displays  almost 
endless  specimens,  and  all  are  in  relief  and  enriched  with  gold  and  color,  chiefly 
blue  and  red.  The  religious  cycles  and  symbolic  figures  of  the  Byzantine  are 
excluded.  Mere  curves  and  angles  or  interlacings  were  now  to  bear  the  chief 
burden  of  a  design,  but  distinguished  by  a  variety  of  color.  The  curves,  how- 
ever, very  naturally  fell  into  standard  forms  and  floral  shapes,  and  the  lines 
and  angles  were  soon  developed  into  a  very  characteristic  species  of  tracery,  or 
interlaid  strap-work,  very  agreeably  diversified  by  the  ornamental  introduction 

88 


594: 


ARCHITECTURAL  DRAWING. 


of  the  inscriptions,  which  last  custom  of  elaborating  inscriptions  with  their 
designs  was  peculiarly  Saracenic.  Although  flowers  were  not  palpably  admit- 
ted, yet  the  great  mass  of  the  minor  details  of  Saracenic  designs  are  composed 
of  flower  forms  disguised — the  very  inscriptions  are  sometimes  thus  grouped  as 
flowers  ;  still,  no  actual  flower  ever  occurs,  as  the  exclusion  of  all  natural  im- 
ages is  fundamental  to  the  style  in  its  purity. 

All  the  symbolic  elements  of  the  Byzantine  are  continued  in  the  Gothic. 
Ornamentally,  the  Gothic  is  the  geometrical  and  pointed  element  elaborated  to 
the  utmost  ;  its  only  peculiarities  are  its  combinations  of  details  ;  at  first  the 
conventional  and  geometrical  prevailing,  and  afterward  these  combined  with 
the  elaboration  of  natural  objects  in  its  decoration.  The  most  striking  feature 
of  all  Gothic  work  is  the  wonderful  elaboration  of  its  geometric  tracery ;  vesi- 
cas,  trefoils,  quatrefoils,  cinquefoils,  and  an  infinity  of  geometric  varieties  be- 
sides. The  tracery  is  so  paramount  a  characteristic  that  the  three  English 
varieties,  the  early  English,  the  decorated,  and  the  perpendicular,  and  the 
French  flamboyant,  are  distinguished  almost  exclusively  by  this  feature.  (See 
Figs.  1360  to  1364.) 

The  ornamental  moldings  used  in  the  decorative  details  are  numerous, 
among  which  the  more  common  is  the  chevron  or  zigzag  (Fig.  1423),  simple 


FIG.  1423. 


FIG.  1424. 


FIG.  1425. 


FIG.  1426. 


FIG.  1428. 


FIG.  1429. 


FIG.  1430. 


as  the  indented,  or  du- 
plicated, triplicated,  or 
quadrupled  ;  the  billet, 
the  prismatic  billet,  the 
square  billet,  and  the 

alternate  billet  (Fig.  1424)  ;  the  star  (Fig.  1425),  the  fir-cone  ;  the  cable  (Fig. 
1426)  ;  the  embattled  (Fig.  1427)  ;  the  nail-head  (Fig.  1428),  the  dog-tooth 
(Fig.  1429)  ;  the  ball-flower  (Fig.  1430),  and  the  serpentine  vine-scroll. 

The  crocket,  in  its  earliest  form,  was  the  simple  arrow-head  of  the  episco- 
pal pastoral  staif  ;  subsequently  finished  with  a  trefoil,  and  afterward  still  fur- 
ther enriched.  Figs.  1431  and  1432  are  early  English  crockets  ;  Fig.  1433  a 
decorated  one.  Fig.  1434  is  a  finial  of  the  same  style.  Both  finials  and  crock- 
ets in  detail  display  a  variety  of  forms. 

The  parapets  of  the  early  English  style  are  often  a  simple  horizontal  course, 
supported  by  a  corbel  table,  sometimes  relieved  by  a  series  of  sunk  blank  trefoil- 
headed  panels  ;  sometimes  a  low  embattled  parapet  crowns  the  wall.  In  the 
decorated  style  the  horizontal  parapet  is  sometimes  pierced  with  trefoils,  some- 


ARCHITECTURAL  DRAWING. 


595 


times  with  wavy,  flowing  tracery  (Fig.  1435).     Grotesque  spouts  or  gargoyles 
discharge  the  water  from  the  gutters.     The  parapets  of  the  perpendicular  style 


FIG.  1431. 


FIG.  1433. 


FIG.  1432. 


FIG.  1434. 


are  frequently  embattled  (Fig.  1436),  covered  with  sunk  or  pierced  paneling, 
and  ornamented  with  quatrefoil,  or  small  trefoil-headed  arches  ;   sometimes 


FIG.  1436. 


FIG.  1435. 

not  embattled  but  covered  with  sunk  or  pierced 
quatrefoils  in  circles,  or  with  trefoils  in  triangular 
spaces,  as  in  Fig.  1437. 

Among  the  varieties  of  ornamental  work,  the 
mode  of  covering  small  plain  surfaces  with  diaper- 
ing (Fig.  1438)  was  sometimes  used  ;  the  design 
being  in  exact  accordance  with  the  architectural 


FIG.  1438. 


FIG.  1437. 


K     FIG.  1439 


features  and  details  of  the  style.  The  rose  (Fig.  1439),  the  badge  of  the 
houses  of  York  and  Lancaster,  is  often  met  with  in  the  perpendicular  style  ; 
and  tendrils,  leaves,  and  fruit  of  the  vine  are  carved  in  great  profusion  in  the 


596 


ARCHITECTURAL  DRAWING. 


hollows  of  rich  cornice  moldings,  especially  on  screen-work  in  the  interior  of  a 
church.     Fig.  1440,  in  its  original  type  a  Byzantine  ornament,  an  alternate 

lily  and  cross,  is  a  common  finish  to  the  cor- 
nice of  rich  screen-work  in  the  latest  Gothic, 
and  is  known  under  the  name  of  the  Tudor 
flower. 

Sculptured  foliage  (Figs.  1441  to  1446) 
is  much  used  in  capitals,  brackets,  corbels, 
bosses,  and  crockets.     Among  the  forms  of 
FIG.  1440  foliage  the  trefoil  is  most  predominant. 


FIG.  1441. 


FIG.  1442. 


FIG.  1443. 


FIG.  1444. 


FIG.  1445. 


FIG.  1446. 


FIG 


1447. 


The  Ornaments  of  the  Renais- 
sance.— The  term  Renaissance  is 
used  in  a  double  sense  ;  in  a  general 
sense  implying  the  revival  of  art, 
and  specially  signifying  a  peculiar 
style  of  ornament.  It  is  also  some- 
times, in  a  very  confined  sense,  ap- 
plied in  reference  to  ornament  of 
the  style  of  Benvenuto  Cellini ;  or, 
as  it  is  sometimes  designated,  the 
Henry  II  (of  France)  style. 

The  mixture  of  various  elements 
is  one  of  the  essentials  of  this  style. 
These  elements  are  the  classical  or- 
naments ;  unnatural  and  natural 
flowers  and  foliage  ;  men  and  ani- 
mals, natural  and  grotesque  ;  car- 


ARCHITECTURAL  DRAWING. 


597 


touches,  or  pierced  and  scrolled  shields,  in  great  prominence  ;  tracery  inde- 
pendent, and  developed  from  the  scrolls  of  the  cartouches ;  and  jewel  forms 
(Fig.  1447  and  1448). 

The  Elizabethan  is  a  partial  elaboration  of  the  same  style  ;  the  present  Eliza- 
bethan exhibits  a  very  striking  preponderance  of  strap  and  shield  work ;  but 


FIG.  IMS. 

the  earlier  is  much  nearer  allied  to  the  Continental  styles  of  the  time,  classical 
ornaments  but  rude  in  detail,  occasional  scroll  and  arabesque  work,  and  strap- 
work,  holding  a  much  more  prominent  place  than  the  pierced  or  scrolled 
shields.  Fig.  1449  is  an 
example  of  the  style  from 
the  old  guard  chamber, 
'Westminster.  *~7^rjymx^\.  ^^  ~w  iv  (<r~$  t\m/s*\> . 


FIG.  1449. 


FIG.  1450. 


FIG.  1451. 


Of  the  earliest  and  transition  styles  of  Eenaissance  ornament  are  the  Tri- 
cento  and  the  Quatrecento  ;  the  great  features  of  the  first  are  its  intricate  tra- 
cery and  delicate  scroll-work  of  conventional  foliage,  the  style  being  but  a  slight 


598 


ARCHITECTURAL  DRAWING. 


remove  from  the  Byzantine  and  Saracenic ;  of  the  second,  elaborate  natural 
imitations  of  fruit,  flowers,  birds,  or  animals  (Fig.  1450),  all  disposed  simply 
with  a  view  to  the  ornamental  ;  also  occasional  cartouches,  or  scrolled  shield- 
work. 

The  Renaissance  is  something  more  approximative  to  a  combination  of  pre- 
vious styles  than  a  revival  of  any  in  particular,  developed  solely  on  aesthetic 
principles,  from  a  love  of  the  forms  and  harmonies  themselves,  as  varieties  of 
effect  and  arrangements  of  beauty,  not  because  they  had  any  particular  signi- 
fication, or  from  any  superstitious  attachment  to  them  as  heirlooms. 

Fig.  1451  is  an  example  of  ornament  in  the  Cinquecento  style.  The  ara- 
besque scroll-work  is  the  most  prominent  feature  of  the  Cinquecento,  and  with 
this  in  its  elements,  it  combines  every  other  feature  of  classical  art,  with  the 
unlimited  choice  of  natural  and  conventional  imitations  from  the  entire  animal 
and  vegetable  kingdom,  both  arbitrarily  disposed  and  combined.  Absolute 
works  of  art,  such  as  vases  and  implements,  and  instruments  of  all  kinds,  are 
prominent  elements  of  the  Cinquecento  arabesque,  but  cartouches  and  strap- 
work  wholly  disappear  from  the  best  examples.  Another  chief  feature  of  the 

Cinquecento  is  the  admirable  play  of  color  in 
its  arabesques  and  scrolls  ;  and  it  is  worthy  of 
note  that  the  three  secondary  colors,  orange, 
green,  and  purple,  perform  the  chief  parts  in 
all  the  colored  decorations. 

Fig.  1452  is  an  example  of  the  Louis  Qua- 
torze  style  of  ornament.  The  great  medium 
of  this  style  was  gilt  stucco-work,  and  this 
absence  of  color  seems  to  have  led  to  its  most 
striking  characteristic,  infinite  play  of  light, 
of  shade  ;  color,  or  mere  beauty  of  form  in 
detail,  having  no  part  in  it  whatever.  Flat 
surfaces  are  not  admitted  ;  all  are  concave  or 
convex  :  this  constant  varying  of  the  surface 
gives  every  point  of  view  its  high  lights  and 
brilliant  contrasts. 

The  Louis  Quinze  style  differs  from  that 
of  Louis  Quatorze  chiefly  in  its  absence  of 
symmetry  ;  in  many  of  its  examples  it  is  an 

almost  random  dispersion  of  the  scroll  and  shell,  mixed  only  with  that  peculiar 
crimping  of  shell-work,  the  coquillage. 

The  ornaments  of  which  we  have  thus  given  examples  are,  in  general,  ap- 
plied to  interior  decorations,  to  friezes,  pilasters,  panels,  architraves,  the  faces 
and  soffits  of  arches,  ceilings,  etc.,  to  furniture,  and  to  art-manufactures  in 
general.  For  exteriors  these  ornaments  are  sparingly  applied  ;  shield  and  scroll 
work,  of  the  later  Elizabethan  or  Renaissance  style,  is  sometimes  used,  but 
very  seldom  tracery. 

Principles  of  Design. — Professedly  treating  of  architecture  only  in  its  most 
mechanical  phase  of  drawing,  the  history  of  it  as  an  art,  and  the  distinctions 
of  styles,  have  been  but  briefly  treated.  To  one  anxious  to  acquire  knowledge- 


FIG.  1452. 


ARCHITECTURAL  DRAWING.  599 

in  this  department  we  refer,  as  the  very  best  compendium  within  our  knowl- 
edge, to  Ferguson's  "  Hand-Book  of  Architecture."  The  study  of  this  work 
will  give  direction  to  a  person's  observation,  but,  without  referring  to  actual 
examples,  mere  reading  will  be  of  little  use.  Drawings  give  general  ideas  of 
the  character  of  buildings,  but  no  idea  of  size  or  of  the  surroundings  of  a 
building.  Many  a  weak  design,  especially  in  cast-iron  buildings,  acquires  a 
sort  of  strength  by  the  number  of  its  repetitions,  giving  an  idea  of  extent ;  and 
many  a  beautiful  design  on  paper  has  failed  in  its  execution,  being  dwarfed  by 
its  surroundings.  With  regard  to  the  style  of  a  building,  there  are  none  of  the 
ancient  styles  in  their  purity  adapted  to  present  requirements  ;  our  churches 
and  theatres  are  more  for  the  gratification  of  the  ear  than  the  eye,  and  the 
comforts  of  our  domestic  architecture,  and  the  requirements  of  our  stores  and 
warehouses,  are  almost  the  growth  of  the  present  century.  For  a  design,  look 
first  to  the  requirements  of  the  structure,  the  purposes  to  which  it  is  to  be 
applied  ;  sketch  the  plan  first,  arrange  the  divisions  of  rooms,  the  openings  for 
doors  and  windows,  construct  the  sections,  and  then  the  elevations,  first  in 
plain  outline  ;  modify  each  by  the  exigencies  of  construction. 

"  Construction,  including  in  the  term  the  disposition  of  a  building  in  ref- 
erence to  its  uses,  is  by  some  supposed  to  be  the  common  part  of  the  art  of 
architecture,  but  it  is  really  the  bone,  muscle,  and  nerve  of  architecture,  and  the 
arts  of  construction  are  those  to  which  the  true  architect  will  look,  rather  than 
to  rules  and  examples,  for  the  means  of  producing  two  at  least  of  the  three 
essential  conditions  of  building  well,  commodity,  firmness,  and  delight,  which 
conditions  have  been  aptly  said  to  be  the  end  of  architecture  as  of  all  creative 
arts. 

"  The  two  great  principles  of  the  art  are  :  First,  that  there  should  be  no 
features  about  a  building  which  are  not  necessary  for  convenience,  construc- 
tion, or  propriety ;  second,  that  all  ornament  should  consist  of  enrichment  of 
the  essential  construction  of  the  building. 

"  The  neglect  of  these  two  rules  is  the  cause  of  all  the  bad  architecture  of  the 

o 

present  time.  Architectural  features  are  continually  tacked  on  buildings  with 
which  they  have  no  connection,  merely  for  the  sake  of  what  is  termed  effect, 
and  ornaments  are  continually  constructed  instead  of  forming  the  decoration 
of  construction  to  which  in  good  taste  they  should  always  be  subservient.  The 
taste  of  the  artist  ought  to  be  held  merely  ancillary  to  truthful  disposition  for 
structure  and  service.  The  soundest  construction  is  the  most  apt  in  the  pro- 
duction or  the  reproduction,  it  may  be,  of  real  art.  The  Eddystone  Lighthouse 
is  well  adapted  to  its  uses  ;  it  is  commodious,  firm  and  stable  almost  to  a  mira- 
cle, and  its  form  is  as  beautiful  in  outline  to  the  delight  of  the  eye,  as  it  is  well 
adapted  to  break  and  mitigate  the  force  of  the  sea  in  defense  of  its  own  struct- 
ure. The  Great  Exhibition  Building  of  1851  was  most  commodious  for  the 
purposes  of  an  exhibition,  firm  enough  for  the  temporary  purpose  required  of 
it,  and  there  was  delight  in  the  simplicity  and  truth  of  its  combinations  ;  and 
all  this  may  be  said  to  have  grown  out  of  propriety  of  construction,  as  applied 
to  the  material,  cast-iron.  The  use  of  unfitting  material,  or  fitting  material 
inappropriately,  leads  almost  entirely  to  incommodiousness,  infirmity,  and 
offense,  or  some  of  them. 


600  ARCHITECTURAL  DRAWING. 

"  Out  of  truth  in  structure,  and  that  structure  of  a  very  inartificial  sort, 
grow  the  beautiful  forms  of  the  admirable  proportions  found  in  the  works  of 
the  Greeks  ;  and  out  of  truth  in  structure,  with  the  strictest  regard  to  the 
necessities  of  the  composition  and  of  the  material  employed,  and  that  structure 
as  full  of  artifice  as  the  artifice  employed  is  of  truth  and  simplicity,  grew  the 
classical  works  vulgarly  called  Gothic,  but  now  characteristically  designated  as 
Pointed,  from  the  arch  which  is  the  basis  of  the  style.  Structural  untruth  is 
not  to  be  justified  by  authority  ;  neither  Sir  Christopher  Wren,  nor  the  Athe- 
nian exemplars  of  Doric  or  Ionic  in  the  Propylaeum  and  in  the  Minerva  Polias, 
with  their  irregular  and  inordinately  wide  intercolumniation,  can  persuade 
even  the  untutored  eye  to  accept  weakness  for  strength,  or  what  is  false  for 
truth. 

"  The  Greek  examples  offer  the  most  beautiful  forms  for  moldings,  and  the 
Grecian  mode  of  enriching  them  is  unsurpassed.  It  should  be  borne  in  mind 
that  the  object  in  architectural  enrichment  is  not  to  show  ornament,  but  to 
enrich  the  surface  by  producing  an  effective  and  pleasing  variety  of  light  and 
shade  ;  but  still,  although  ornament  should  be  a  secondary  consideration,  it 
will  develop  itself,  and  therefore  should  be  of  elegant  form  and  composition." 

We  have  quoted  thus  at  some  length  from  the  article  "  Architecture," 
"  Encyclopaedia  Britannica,"  because  with  many  authority  is  necessary,  and 
they  distrust  their  own  powers  of  observation  and  analysis ;  all  must  feel  the 
truth  of  the  above,  but  in  practice  it  is  very  little  appreciated  or  carried  out. 
The  present  taste  in  architecture,  as  in  the  theatre,  is  for  the  spectacular  ; 
breadth  or  dignity  of  effect  is  not  popular  ;  edifices  are  not  only  covered  with, 
but  built  up  in  ornament ;  and  construction  is  but  secondary.  The  French, 
having  a  building-stone  that  is  very  easily  worked,  cut  merely  the  joints,  leav 
ing  the  rough  outer  surface  to  be  worked  after  it  is  laid  ;  chopping  out  mold- 
ings and  ornaments  almost  as  readily  as  though  it  were  in  plaster,  and  the  sur- 
face when  finished  is  covered  with  enrichments  in  low  relief.  The  fashion  thus 
set  is  imitated  in  this  country  at  immense  cost,  in  the  most  unfitting  materials, 
marble  and  granite.  Our  architectural  buildings  express  fitly  our  condition — 
a  rich  country,  recent  and  easily  acquired  wealth,  and  a  desire  and  rivalry  to 
exhibit  it,  or  a  display  as  a  means  of  advertising,  and  in  this  truth  of  expression 
will  have  an  archaeological  interest ;  although  it  does  not  contribute  much  to 
present  excellence  in  construction,  it  still  has  this  value,  that  the  architect  or 
constructor  need  be  governed  by  no  rules  or  principles — he  can  make  experi- 
ments on  a  pretty  extensive  scale,  and  out  of  much  bad  construction  even  forms 
and  ornament  may  spring  up  which  will  stand  the  test  of  time,  and  form  a 
nucleus  of  a  new  style  adapted  to  the  present  wants. 

Cast-iron  as  a  building  material,  with  the  exception  of  exhibition-buildings, 
has  seldom  been  treated  distinctively  ;  buildings  erected  with  it  have  been 
copies  of  those  in  stone,  and  have  been  even  imitated  in  color.  For  the  first 
story  of  stores,  where  space  is  necessary  for  light  and  the  exhibition  of  wares, 
cast-iron  columns  are  almost  invariably  used,  but  are  objected  to  architecturally, 
that  they  look  too  weak  for  the  support  of  the  piles  of  brick  and  stone  above 
them.  The  objection  should  not  be  to  the  use,  but  that  the  truth  of  the  ade- 
quate strength  of  the  cast-iron  is  not  conveyed  by  the  form  or  color.  "No  one 


ARCHITECTURAL   DRAWING.  601 

objects  that  the  ankles  of  Atlas  look  too  light  to  support  the  massive  figure  and 
globe,  or  wishes  him  seated  to  give  the  idea  of  stability ;  so  if  the  columns  and 
lintels  were  some  other  form  than  Greek  or  Koman  with  immense  inter- 
columniations,  and  colored  fitly,  the  appearance  of  weakness  would  be  entirely 
lost  sight  of. 

In  conclusion,  the  draughtsman  should  be  conversant  with  classic  and  later 
styles,  still,  as  he  must  design  to  suit  the  necessities  of  the  times,  and  the 
requirements  of  present  tastes  and  fashions  of  buildings,  he  should  keep  him- 
self posted  on  what  is  being  done,  and  he  will  find  it  very  convenient  to  have  a 
scrap-book  of  cuts  from  which  to  draw  parts  of  a  design,  and  afford  him  ready 
means  of  combinations.  He  will  find  much  in  illustrated  magazines  and  news- 
papers, many  cuts  unpromising  as  a  whole,  yet  fruitful  in  suggestions  of  parts  ; 
many  an  agreeable  outline  illy  filled  up  ;  many  that  are  only  valuable  as  showing 
dimensions  requisite  for  certain  uses.  But  the  larger  the  collection  the  better 
for  the  draughtsman  ;  it  will  save  time  to  know,  as  far  as  possible,  what  has 
been  done,  that  he  may  judge  what  forms  and  proportions  it  will  be  best  for 
him  to  use,  and  what  to  avoid. 

It  has  been  our  practice  to  select,  from  papers  and  magazines,  cuts  which 
we  considered  of  value,  and  arrange  them  in  scrap-books  with  appropriate 
headings.  In  the  Appendix  a  few  pages  of  "  scraps  v  are  given  as  illustrations. 


PERSPECTIVE  DRAWING. 


o 


FIG.  1453. 


THE  science  of  Perspective  is  the  representation  by  geometrical  rules,  upon 
a  plane  surface,  of  objects  as  they  appear  to  the  eye,  from  any  point  of  view. 

All  the  points  of  the  surface  of  a  body 
are  visible  by  means  of  luminous  rays  pro- 
ceeding from  these  points  to  the  eye.  Thus, 
let  the  line  A  B  (Fig.  1453)  be  placed  before 
the  eye,  C,  the  lines  drawn  from  the  differ- 
ent points  1,  2,  3,  4,  etc.,  represent  the 
visual  rays  emanating  from  each  of  these 
points.  It  is  easy  to  understand  that,  if  in 
the  place  of  a  line  a  surface  is  substituted, 
the  result  will  be  a  pyramid  of  rays. 

Let  A  B  (Fig.  1454)  be  a  straight  line, 
and  let  the  globe  of  the  eye  be  represented 

by  a  circle,  and  its  pupil  by  the  point  C.     The  ray  emanating  from  A,  enter- 
ing through  C,  will  proceed  to  the  retina  of  the  eye,  and  be  depicted  at  a. 

And  as  it  follows  that 
all  the  points  of  A  B 
will  send  rays,  enter- 
ing the  eye  through  C, 
the  whole  image  of  A 
B  will  be  depicted  on 
the  retina  of  the  eye 
in  a  curved  line  a  3  b. 
Conceive  the  line  A  B 
moved  to  a  greater  dis- 
tance from  the  eye,  and 
placed  at  A'  B',  then 
the  optic  angle  will  be 
reduced,  and  the  image 
a'  3  b'  will  be  less  than 
before  ;  and  as  our  vis- 

Fio.  1454.  ual   sensations   arc    in 

proportion  to  the  mag- 
nitude of  the  image  painted  on  the  retina,  it  may  be  concluded  that  the  more 
distant  an  object  is  from  the  eye  the  smaller  the  angle  under  which  it  is  seen 
becomes,  and,  consequently,  the  less  it  appears. 


PERSPECTIVE  DRAWING.  603 

Observation  has  rendered  it  evident  that  the  greatest  angle  under  which 
one  or  more  objects  can  be  distinctly  seen  is  one  of  90°.  If  between  the  ob- 
ject and  the  eye  there  be  interposed  a  transparent  plane  (such  as  one  of  glass, 
m  n),  the  intersections  of  this  plane  with  the  visual  rays  are  termed  perspectives 
of  the  points  from  which  the  rays  emanate.  Thus  a  is  the  perspective  of  A, 
b  of  B,  and  so  on  of  all  the  intermediate  points  ;  but,  as  two  points  determine 
the  length  of  a  straight  line,  it  follows  that  a  b  is  the  perspective  of  A  B,  and 
a"  I"  the  perspective  of  A'  B'. 

It  is  evident  from  the  figure  that  objects  appear  larger  or  smaller  according 
to  the  angle  under  which  they  are  viewed  ;  and  further,  that  objects  of  une- 
qual size  may  appear  equal  if  seen  under  the  same  angle.  For,  draw  fg,  and 
its  perspective  will  be  found  to  be  the  same  as  that  of  A'  B'. 

It  follows  also  that  a  line  near  the  eye  may  be  viewed  under  an  angle  much 
greater  than  a  line  of  greater  dimensions  but  more  distant,  and  hence  a  little 
object  may  appear  to  be  much  greater  than  a  similar  object  of  larger  dimen- 
sions. Since,  therefore,  unequally  sized  objects  may  appear  equal  in  size,  and 
equally  sized  objects  unequal,  and  since  objects  are  not  seen  as  they  are  in 
reality,  but  as  they  appear  under  certain  conditions,  perspective  may  be  defined 
to  be  a  science  which  affords  the  means  of  representing,  on  any  surface  what- 
ever, objects  such  as  they  appear  when  seen  from  a  given  point  of  view.  It  is 
divided  into  two  branches,  the  one  called  linear  perspective,  occupying  itself 
with  the  delineation  of  the  contours  of  bodies,  the  other  called  aerial  perspec- 
tive, with  the  gradations  of  colors  produced  by  distance.  It  is  the  former  of 
these  only  that  is  proposed  here  to  be  discussed. 

The  perspective  of  objects,  then,  is  obtained  by  the  intersection  of  the  rays 
which  emanate  from  them  to  the  eye,  by  a  plane  or  other  surface  (which  is 
called  the  picture),  situated  between  the  eye  and  the  objects. 

From  the  explanation  and  definition  just  given,  it  is  easy  to  conceive  that 
linear  perspective  is  in  reality  the  problem  of  constructing  the  section,  by  a  sur- 
face of  some  kind,  of  a  pyramid  of  rays  of  which  the  summit  and  the  base  are 
given.  The  eye  is  the  summit,  the  base  may  be  regarded  as  the  whole  visible 
extent  of  the  object  or  objects  to  be  represented,  and  the  intersecting  surface  is 
the  picture. 

A  good  idea  of  this  will  be  obtained  by  supposing  the  picture  to  be  a  trans- 
parent plane,  through  which  the  object  may  be  viewed,  and  on  which  it  may 
be  depicted. 

In  addition  to  the  vertical  and  horizontal  planes  with  which  we  are  familiar 
in  the  operations  of  projection,  several  auxiliary  planes  are  employed  in  perspec- 
tive, and  particularly  the  four  following  : 

1.  The  horizontal  plane  A  B  (Fig.  1455),  on  which  the  spectator  and  the 
object  viewed  are  supposed  to  stand,  for  convenience  supposed  perfectly  level, 
is  termed  the  ground  plane. 

2.  The  plane  M  N,  which  has  been  considered  as  a  transparent  plane  placed 
in  front  of  the  spectator,  on  which  the  objects  are  delineated,  is  called  the  plane 
of  projection  or  the  plane  of  the  picture.     The  intersection  M  M  of  the  first 
and  second  planes  is  called  the  line  of  projection,  the  ground,  or  base  line  of 
the  picture. 


604 


PERSPECTIVE  DRAWING. 


3.  The  plane  E  F  passing  horizontally  through  the  eye  of  the  spectator,  and 
cutting  the  plane  of  the  picture  at  right  angles,  is  called  the  horizontal  plane, 
and  its  intersection  at  D  D  with  the  plane  of  the  picture  is  called  the  horizon 
line,  the  horizon  of  the  picture,  or  simply  the  horizon. 

4.  The  plane  S  T  passing  vertically  through  the  eye  of  the  spectator,  and 
cutting  each  of  the  other  planes  at  a  right  angle,  is  called  the  central  plane. 

Point  of  view,  or  point  of  sight,  is  the  point  where  the  eye  is  supposed  to 
be  placed  to  view  the  object,  as  at  0,  and  is  the  vertex  of  the  optical  pyramid. 
Its  projection  on  the  ground  plane  S  is  termed  the  station  point. 

The  projection  of  any  point  on  the  ground  plane  is  called  the  seat  of  that 
point. 

Center  of  view  (commonly,  though  erroneously,  called  the  point  of  sight), 
is  the  point  V  where  the  central  vertical  line  intersects  the  horizon  line  ;  a  line 
drawn  from  this  point  to  the  eye  would  be  in  every  way  perpendicular  to  the 
plane  of  the  picture. 

Points  of  distance  are  points  on  the  horizontal  line  as  remote  from  the 
-centre  of  view  as  the  eye. 


M 


M 


Fia.  1455. 


Vanishing  points  are  points  in  a  picture  to  which  all  lines  converge  that 
in  the  original  object  are  parallel  to  each  other. 

Parallel  Perspective. — An  object  is  said  to  be  seen  in  parallel  perspective 
when  one  of  its  sides  is  parallel  to  the  plane  of  the  picture. 

Angular  Perspective. — An  object  is  said  to  be  seen  in  angular  perspective 
when  none  of  its  sides  are  parallel  to  the  picture. 

To  find  the  perspective  of  points,  as  the  points  m,  s  (Fig.  1456),  in  the  ground 
plane,  the  same  letters  designating  similar  planes  and  points  as  in  Fig.  1455. 
From  the  point  m  draw  a  line  to  the  point  of  sight  C,  and  also  to  the  station 
point  S  ;  at  the  intersection  of  the  line  m  S  with  the  base  line  MS',  erect  a  per- 
pendicular cutting  the  line  m  C,  the  intersection  m'  will  be  the  perspective 
projection  of  the  point  m,  on  the  plane  of  the  picture  M  V.  The  point  s  being 
in  the  central  plane,  its  projection  must  be  in  the  intersection  of  that  plane  by 
the  plane  of  the  picture,  at  the  point  s'  the  intersection  of  the  central  vertical 
line  by  the  line  s  C. 


PERSPECTIVE  DKAW 

In  the  same  way  find  the  perspective  h'  m'  of 
when  an  original  line  is  parallel  or  perpendicular 
perspective  of  that  line  will  also  be  parallel  or  perpen 


605- : 


e  h  m,  and  we  find  that 
base  of  the  picture,  the 
to  it. 


FIG.  1456. 

Fig.  1457.  Draw  the  diagonals  M  s'  and  m  S',  project  as  in  the  preceding 
figure  the  points  m  and  s  into  the  plane  of  the  picture,  draw  M  m'  M  S',  and 
S'  m' ;  now,  since  m  and  M  are  the  extremities  of  a  line  perpendicular  to  the 
plane  of  the  picture,  the  line  m'  M  must  be  the  projection  of  this  line  on  the 
plane  of  the  picture,  and  if  this  line  be  extended  it  will  pass  through  V,  which 
may  be  demonstrated  of  all  lines  perpendicular  to  the  plane  of  the  picture  ; 
hence  the  perspective  direction  of  lines  perpendicular  to  the  picture  is  to  the 
center  of  view. 

v/ 


FIG.  1457. 


If  the  line  m' S'  be  extended  it  will  pass  through  the  point  D,  and  if  M  s' 
be  extended  it  will  pass  through  a  point  in  the  line  of  the  horizon  at  a  distance 
from  V  equal  to  V  D  ;  by  construction  D  V  has  been  made  equal  to  V  C,  and 


606 


PERSPECTIVE  DRAWING. 


as  this  demonstration  is  applicable  to  other  similar  lines,  and  since  M  m  s  S'  is 
a  square  ;  hence  the  perspective  direction  of  all  lines,  making  an  angle  0/45° 
with  the  plane  of  the  picture,  is  toward  the  point  of  distance. 

Having  thus  illustrated  the  rules  of  parallel  perspective,  we  now  proceed  to 


apply  them  to  the  drawing  of  a  square  and  cube  (Fig.  1458).  The  same  letters 
are  employed  in  similar  positions  as  in  preceding  figures. 

It  is  necessary  to  premise  that  the  student  should  draw  these  examples  at 
least  three  times  the  size  of  those  in  Fig.  1458. 

Let  A  and  B  (Fig.  1458)  represent  the  plan,  or  situation  upon  the  ground, 


PERSPECTIVE  DRAWING. 


607 


of  two  squares,  of  which  a  perspective  representation  is  required.  First  draw 
the  line  M  M,  which  represents  the  base  line  of  the  picture  ;  make  S  the  station 
point  or  place  of  the  observer,  and  draw  lines  or  rays  from  all  visible  angles  of 
the  squares,  to  S  ;  then  draw  the  lines  S  M,  parallel  to  the  diagonal  lines  of  the 
squares.  Now  draw  M'  M'  parallel  to  M  M  representing  the  base  line  of  the 
picture  in  elevation  ;  then  draw  S'  V,  the  vertical  line  immediately  opposite  the 
eye  ;  let  the  distance,  S'  Y,  be  the  height  of  the  eye  from  the  ground,  and  draw 
D  D  the  horizontal  line  ;  V  being  the  center  of  view  ;  let  fall  perpendicular 
lines  from  the  angles  a  and  b  of  the  plan  of  the  square  A,  and  also  from  the 
point  c,  where  the  ray  from  the  angle  e  intersects  the  base  line,  M  M  ;  from  a1 
and  V  draw  lines  to  the  center  of  view,  V  ;  and  e'  where  the  perpendicular  line 
from  c  intersects  the  line  V  V,  will  give  the  apparent  or  perspective  width  br  e' 
of  the  side  b  e  ;  from  e'  draw  a  line  parallel  to  a'  b',  and  the  perspective  repre- 
sentation of  the  nearest  square  A  is  complete.  In  order  to  prove  the  accuracy 
of  this  performance,  it  is  necessary  to  try  if  the  diagonal  lines,  a'  e',  and  b'f, 
incline  respectively  to  the  points  of  distance,  D  D,  on  the  horizontal  line  :  if 
so,  it  is  correct.  The  square  B  is  drawn  in  precisely  the  same  manner,  and  will 
be  easily  understood  by  observing  the  example. 

The  plans  of  the  two  cubes  C  and  D  are  the  same  as  the  plans  of  the 
squares  A  and  B.  As  neither  of  these  cubes  appears  to  touch  the  plane  of  the 
picture  M  M,  it  will  be  necessary  to  imagine  the  sides  I  g,  and  Jc  h,  to  be  con- 
tinued until  they  do  so  ;  now  draw  down  perpendicular  lines  from  where  the 
continuations  of  these  sides  intersect  the  base  line,  and  set  off  on  them  from 
the  line  M'  M',  the  height  of  the  cube,  as  1 — 2  which  is  the  same  as  the  width, 
and  complete  the  square  shown  by  the  dotted  lines  ;  from  all  four  angles  of  this 
square  draw  lines  to  the  center  of  view — this  will  give  the  representation  of 
four  lines  at  right  angles  with  the  picture  carried  on  as  far  as  it  would  be  pos- 


sible  to  see  them  ;  then  it  only  remains  to  cut  off  the  required  perspective 
widths  of  the  cubes,  by  the  perpendicular  lines,  from  the  intersection  of  the 
visual  rays  with  the  plane  of  the  picture  :  the  completion  of  this  problem  will 
be  very  easy,  if  the  drawing  of  the  squares  is  well  understood. 

In  such  simple  objects  as  these  it  will  not  be   necessary  to  draw  a  plan 


608 


PERSPECTIVE  DRAWING. 


when  one  side  is  parallel  to  the  picture,  and  dimensions  are  known.  In  Fig. 
1459,  the  same  objects  as  those  in  Fig.  1458  are  drawn  without  a  plan  thus  : 

Draw  the  ground  line  M  M,  then  the  vertical  line  S'  V,  and  the  horizontal 
line  D  D,  at  the  height  of  the  eye  ;  making  D  D  the  same  distance  on  each 
side  of  V  that  the  eye  is  from  the  transparent  plane  ;  for  drawing  the  squares, 
mark  off  from  S'  to  b',  on  the  ground  line,  the  distance  that  the  square  is  on 
one  side  of  the  observer  ;  let  b'  a'  be  the  length  of  one  side  of  the  square  ;  from 
b'  and  a'  draw  lines  to  V,  which  represent  the  sides  of  the  square  carried  on  in- 
definitely ;  to  cut  off  the  required  perspective  width  of  the  side  b'  er  of  the 
square,  lay  off  the  width,  a'  b',  from  bf  to  p,  then  draw  from  p  to  D  on  the  left 
and  the  point  e'  where  the  line  Dp  intersects  b'  V  will  give  the  apparent  width 
required  ;  then  draw/'  e'  parallel  to  a'  b',  and  the  square  is  complete  :  this  may 
be  proved  in  the  same  way  as  in  Fig.  1458.  The  further  square  may  be  obtained 
in  a  similar  manner,  setting  off  the  distance  between  the  squares  from  p  to  q, 
and  the  width  of  the  square  beyond  that,  and  drawing  lines  to  D  as  before  : 
some  of  the  lines  in  this  plate  are  not  continued  to  the  ground  line,  in  order  to 
avoid  confusion.  Proceed  with  the  cubes  by  the  same  rule.  Let  1,  2,  3,  4,  be 
the  size  of  one  side  of  the  cube  if  continued  until  touching  the  picture  ;  from 
these  points  draw  rays  to  V  ;  from  3  to  t  set  off  the  distance  the  cube  is  from 
the  picture,  and  from  t  to  r,  the  width  of  the  cube  ;  draw  from  these  points  to 
D  on  the  right,  and  their  intersections  of  the  line  3  V  in  m,  o,  will  give  the 
perspective  width  and  position  of  that  side  of  the  cube  ;  then  finish  the  cube 
as  in  the  figure.  The  operation  of  drawing  the  other  cube  is  similar,  and  easy 
to  be  understood. 

From  the  drawing  of  a  square  in  parallel  perspective,  we  deduce  rules  for 
the  construction  of  a  scale  in  perspective.  Let  D  M  M  D  (Fig.  1460)  be  the 
plane  of  the  picture,  the  same  letters  of  reference  being  used  as  in  the  preceding 


M 


figures.  From  S'  lay  off  the  distance  o  S'  equal  to  some  unit  of  measure,  as 
may  be  most  convenient ;  from  o  draw  the  diagonal  to  D  the  point  of  distance  ; 
now  draw  1 1'  parallel  to  the  ground  line  M  M,  again  draw  from  1'  the  diagonal 
I'D,  and  lay  off  the  parallel  2  2',  proceed  in  the  same  way  with  the  diagonal 
2'  D  and  the  parallel  3  3',  and  extend  the  construction  as  far  as  may  be  neces- 


PERSPECTIVE  DRAWING. 


609 


sary.  It  is  evident  o  S'  1 1',  1'  1  2  2',  2'  2  3  3'  are  the  perspective  projections  of 
equal  squares,  and  therefore  o  S',  1  1',  2  2'  3  3',  etc.,  and  S'  1,  1  2,  2  3,  etc.,  are 
equal  to  each  other,  and  that  if  o  S'  is  set  off  to  represent  any  unit  of  measure, 
as  one  foot,  one  yard,  or  ten  feet,  etc. ,  each  of  these  lines  represents  the  same 
distance,  the  one  being  measured  parallel  to  the  base  line,  the  others  perpen- 
dicular to  it.  In  making  a  perspective  drawing  a  scale  thus  drawn  will  be 
found  very  convenient ;  but  as  in  the  center  of  the  picture  it  might  interfere 
with  the  construction  lines  of  the  object  to  be  put  in  perspective,  it  is  better  that 
the  scale  be  transferred  to  the  side  of  the  picture  a  M  o,  the  diagonals  to  be  laid 
off  to  a  point  to  the  right  of  D  equal  to  the  point  of  distance. 

The  scales  thus  projected  are  for  lines  in  the  base  or  ground  plane  ;  for  lines 
perpendicular  to  this  plane  the  following  construction  is  to  be  adopted  :  Upon 
any  point  of  the  base  line  removed  from  S',  as  a  for  instance,  erect  a  perpen- 
dicular, a  d  ;  on  this  line,  lay  off  as  many  of  the  units  o  S'  as  may  be  necessary  ; 
in  this  example  three  have  been  laid  off,  that  is,  a  d  =  3  o  S'.  From  a  and  d 
draw  lines  to  the  center  of  view,  and  extend  the  parallels  1 1',  2  2',  33';  at  the 
intersection  of  these  lines  with  a  V  erect  perpendiculars.  The  portions  com- 
prehended between  the  lines  a  V  and  d  V  will  be  the  perspective  representa- 
tions of  the  line  a  d,  in  planes  at  distances  of  1,  2,  3,  o  S'  from  the  base  line, 
and  as  b,  c,  d  are  laid  off  at  intervals  equal  to  o  S',  by  drawing  the  lines  c  V 
and  b  V  nine  equal  squares  are  constructed,  of  which  the  sides  correspond  to 
the  unit  of  measure  o  S' 

To  determine  the  Perspective  Position  of  any  point  in  the  Ground  Plane. — 
Thus  (Fig.  1461),  to  determine  the  position  of  the  point  p,  which  in  plane  would 
be  six  feet  distant  from  the  plane  of  the  picture,  M  D,  and  ten  feet  from  the 
central  plane,  to  the  left. 

Lay  off  from  S',  to  the  left,  the  distance  a  S',  equal  to  six  feet  on  the  scale 
adopted  ;  draw  the  diagonal  to  the  point  of  distance  D  on  the  right  :  at  its 
intersection  /  with  the  vertical  line  V  S'  draw  a  parallel  to  M  M  ;  lay  off  from 
S',  S'  b  equal  to  ten  feet,  draw  b  V ;  the  intersection  of  this  line  p,  with  the 
parallel  previously  drawn,  will  be  the  position  of  the  point  required. 


D 


By  a  similar  construction  the  position  of  any  point  in  the  ground  plan  may 
be  determined.  It  is  not  necessary  that  the  distances  should  be  expressed  nu- 
merically ;  they  may  be  shown  on  the  plan  and  thence  be  transferred  to  the 
base  line,  and  thrown  into  perspective  by  the  diagonals  and  parallels.  As  the 
intersections  of  the  various  lines  of  the  outlines  of  objects  are  points,  by  pro- 

39 


610 


PERSPECTIVE   DRAWING. 


jecting  perspectively  these  points,  and  afterward  connecting  by  lines,  the  per- 
spective of  any  plane  surface  on  the  ground  plane  may  be  shown. 

If  the  pointy  were  not  in  the  ground  plane,  but  in  a  position  directly  above 
the  ground  plane,  say  five  feet,  then  at  b  erect  a  perpendicular,  and  lay  off 
Z»  V  equal  to  five  feet,  connect  V  V,  at  p  erect  another  perpendicular,  and  its 
intersection  p'  with  the  line  V  V  will  be  the  position  of  the  point  required. 

To  draw  an  Octagon  in  Parallel  Perspective. — Let  A  (Fig.  1462)  represent 
the  plan  of  an  octagon.  Draw  M  M,  S'  V,  and  D  D,  as  before  ;  from  the 
points  M,  a,  b,  c,  draw  rays  to  V.  Set  off  on  MM  from  c  to  the  right  the  dis- 
tances ce,  cd,  cf,  from  which  draw  diagonals  to  D  on  the  left,  and  at  their 
intersection  with  the  ray  c  V,  draw  parallels  e'  g',  d'  h ',  k'  I',  to  the  base  line  ; 
these  points  will  correspond  to  the  angles  on  the  plan.  Now  connect  the  an- 
gles on  the  perspective  view,  in  the  proper  succession,  and  the  perspective  pro- 
jection is  complete. 


It  will  be  observed,  that  in  this  construction  the  plan  has  been  placed  for- 
ward of  the  plane  of  the  picture,  contrary  to  the  position  it  should  occupy, 
which  should  be  the  same  relative  position  back  of  this  plane  ;  but  it  will  be 
found  much  simpler  in  construction  than  if  it  were  placed  as  in  Fig.  1458,  and 
the  points  were  all  projected  to  the  base  line  ;  it  is,  of  course,  equally  correct 
in  its  perspective  projection. 

To  draw  a  Circle  in  Parallel  Perspective. — Let  C  (Fig.  1462)  represent  the 
plan  of  a  circle,  round  which  let  the  square  a  e  c  m  be  described,  two  of  its  sides 
being  parallel  to  the  base  line  M  M  ;  draw  diagonals  across  the  square,  and 
where  these  intersect  the  circumference  of  the  circle  draw  the  lines  bJc  and  dg 
parallel  to  the  base  line,  and  the  lines  on  and  pg  at  right  angles  thereto. 
Draw  also  the  lines/?  and  ch  at  right  angles  to  each  other  through  the  center 


PERSPECTIVE  DRAWING. 


611 


of  the  circle,  project  the  points  a,  o,  I,  p,  m,  to  the  base  and  draw  rays  to  V  ; 
set  off  from  a'  to  the  left  the  distances  a'  a,  a'  b,  a'  c,  a'  d,  a'  e,  and  draw  diago- 
nals to  the  point  of  distance  D  on  the  right ;  at  their  intersection  with  the  line 
a'  V  draw  horizontal  lines,  or  parallels  to  the  base,  and  there  will  be  projected  in 
perspective  the  square  ae  cm,  with  all  the  lines  of  parallels  and  perpendicu- 
lars ;  connect  the  intersections  corresponding  to  the  points  c,  n,  f,  g,  h,  k,  I,  r, 
.and  we  have  the  perspective  projection  of  the  required  circle,  which  will  be  an 
ellipse. 

To  erect  upon  the  octagonal  base  A  an  octagonal  pillar  or  tower.  This  con- 
struction resolves  itself  into  simply  constructing  another  octagon  on  an  upper 
plane,  and  connecting  the  visible  angles  by  perpendiculars  ;  or  perpendiculars 
may  be  erected  at  the  points  M,  a,  Z»,  c,  and  the  heights  of  the  tower  laid  off 
upon  them,  and  from  these  extremities  rays  drawn  to  the  center  of  view  ;  the 
intersection  of  these  rays  by  perpendiculars  from  the  angles  of  the  octagon  be- 
neath will  determine  the  projection  of  the  upper  surface  of  the  pillar ;  repre- 
,sent  in  full  lines  all  visible  outlines,  and  the  projection  is  complete. 

In  the  same  manner  a  pillar  may  be  erected  on  the  circular  base.  If  the 
pillars  be  inclined,  the  first  method  of  projecting  the  upper  outline  on  a  plane 
assumed  at  the  height  of  the  pillar  must  be  adopted. 


VLL 


FIG.  1463. 


To  draw  a  Pyramid  in  Parallel  Perspective. — Let  A  (Fig.  1463)  be  the  plan 
•of  a  pyramid,  the  diagonal  lines  represent  the  angles,  and  their  intersection  the 
vertex  ;  project  the  plan  as  in  previous  examples  of  squares.  Draw  diagonal 
lines  from  M  to  #,  and  a  to  c,  their  intersection  gives  the  perspective  center  of 
the  square  ;  upon  this  point  raise  a  perpendicular  line  which  is  the  axis  of  the 
pyramid ;  draw  a  perpendicular  line  ef,  in  the  center  of  the  line  M  a,  upon 
which  set  up  the  height  of  the  pyramid  ef\  from /draw  a  line  to  V,  and  its 
intersection  of  the  axis  of  the  pyramid  at  d  will  give  the  perspective  height ; 
complete  the  figure  by  drawing  lines  from  rf,  the  apex,  to  M,  a,  b,  the  three 
visible  angles.  The  other  two  pyramids  are  drawn  in  a  similar  manner,  by 
setting  their  distances  from  the  plane  of  the  picture  off  from  a,  on  the  ground 
line  to  the  right,  and  drawing  diagonals  to  the  point  of  distance  on  the  left. 

To  draw  a  Cone  in  Parallel  Perspective. — Let  B  (Fig.  1463)  represent  the 


612  PERSPECTIVE  DRAWING. 

plan  of  a  cone,  apply  the  same  lines  of  construction  as  to  C  (Fig.  1462) ;  and 
draw  the  perspective  view  of  the  circle,  lay  off  the  height  and  finish  precisely 
as  in  the  preceding  case. 

To  draw  a  Square  and  Cube  in  Angular  Perspective. — Let  A  (Fig.  1464) 
be  the  plan  of  the  square,  and  B  the  plan  of  the  cube,  M  M  the  base  or  ground 
line,  and  S  the  station  point.  Draw  M'  M',  and  D  D'  parallel  to  M  M,  the 
one  being  the  ground  line  and  the  other  the  horizon  of  the  plane  of  the  pict- 
ure ;  project  the  point  d  on  MM,  to  d'  on  M'  M'.  It  has  been  shown  in 
parallel  perspective  that  the  vanishing  points  of  diagonals  of  squares  lie  in 
the  points  of  distance  ;  if  through  the  station  point  S,  in  any  of  the  preceding 
figures,  lines  be  drawn  parallel  to  the  diagonals,  they  will  intersect  the  base 
lines  at  distances  from  the  central  plane  equal  to  the  points  of  distance.  In 
like  manner  to  find  the  vanishing  points  of  lines  in  the  ground  planes,  or  in 
planes  parallel  to  the  ground  plane,  inclined  to  the  plane  of  the  picture, 
through  the  station  point  S  draw  lines  parallel  to  the  inclined  lines,  and  pro- 
ject their  intersection  with  the  base  line  to  the  horizon  of  the  picture  ;  thus,  in 
the  present  example,  draw  S  M,  S  M  parallel  to  a  d,  e  h,  and  to  dc,  Jig ;  pro- 
ject their  intersections  M,  M,  with  the  base  line  to  D,  D',  the  horizon  of  the 
picture,  and  D,  D',  will  be  the  vanishing  points  of  all  lines  parallel  to  a  d  and 
d  c.  Draw  d'  D  and  d'  D',  the  perspective  projection  ofda  will  lie  in  the 
former  of  these  lines  and  d  c  in  the  latter.  To  determine  the  perspective  po- 
sition of  the  points  a  and  c,  or  the  length  of  these  lines,  draw  the  rays  a  S  and 
c  S,  project  their  intersection  with  the  base  M  M,  upon  the  lines  d'  D  and  d'  D', 
and  their  intersections  a',  c'  will  be  the  perspective  projection  of  the  points  a 
and  c.  To  complete  the  projection  of  the  square,  draw  the  lines  a'  D'  and  c'  D, 
their  intersection  will  be  the  perspective  projection  of  the  point  b,  and  the 
square  is  complete.  To  prove  the  construction,  draw  the  ray  b  S  and  project 
its  intersection  with  the  base  M  M,  and  if  the  construction  be  correct  it  will 
fall  upon  the  point  b'. 

As  the  cube  is  placed  at  some  distance  from  the  plane  of  the  picture,  it  will 
be  necessary  to  continue  either  eh  or  g  h,  or  both,  till  they  intersect  the  base 
line  M  M  at  n  and  m  ;  drop  perpendiculars  or  project  these  points  upon  M'  M' 
at  n'  and  m' ;  on  these  perpendiculars  set  up  the  height  of  the  cube  m' o  and 
n'  s,  draw  the  lines  m'  D',  o  D',  and  n'  D,  s  D  ;  connect  the  intersections  h'  and 
h"  ;  draw  the  rays  Qe  and  S</,  and  project  their  intersections  with  MM,  to 
g'e' ;  draw  the  lines  e"Df  and  g"  D  ;  if  the  construction  be  correct,  the  projec- 
tion of  the  intersection  of  the  ray  S/  with  the  base  will  fall  upon  /",  and  of 
the  ray  S  li  will  fall  upon  h"  and  h'. 

To  solve  the  Same  Problem  by  a  Different  Construction. — Let  AB  (Fig. 
1464)  be  as  before  the  plans  of  the  square  and  of  the  cube  ;  to  project  them 
perspectively  on  the  plane  of  the  picture  M  D  D'  M  (Fig.  1465). 

From  the  point  M  and  M  (Fig.  1464),  set  off  distances  equal  to  M  S,  M  S,  to 
p  and  p' ;  project  these  points  upon  D  D'  (Fig.  1465),  the  point  p'  (Fig.  1465) 
will  be  that  from  which  any  number  of  parts  may  be  laid  off  on  lines  vanishing 
in  D'  ;  the  point  p  will  be  the  corresponding  point  for  lines  vanishing  in  D. 
These  points  may  be  called  the  points  of  division.  In  parallel  perspective  the 
points  of  distance  were  the  points  of  division,  the  one  for  the  other.  To  illus- 


PERSPECTIVE  DRAWING. 


613 


FIG.  1465. 


614:  PERSPECTIVE  DRAWING. 

trate  their  application  in  the  present  example,  project  the  point  d  (Fig.  1464) 
to  d1  (Fig.  1465),  draw  d'D  and  d'  D',  from  d'  on  either  side  lay  off  a  distance 
d'  i,  df  k  equal  to  the  side  of  the  square  a  d.  Now,  since  p  is  the  division  point 
of  lines  vanishing  in  D,from  i,  draw  the  line  ip,  and  its  intersection  with  d'T> 
cuts  off  a  line  d'  a1  equal  perspectively  to  the  line  d'  i  or  ad  measured  on  the 
base  line.  Again,  since  p'  is  the  division  point  of  lines  vanishing  in  D',  the 
line  k  p'  cuts  off  on  d'  D',  a  line  d'  c'  equal  perspectively  to  the  line  d'  k,  or  a  d 
measured  on  the  base  :  having  a'  d'  c,  the  square  is  completed  by  drawing  the 
lines  c'  b'  toward  D,  and  a'  b'  toward  D'. 

To  construct  the  cube,  project  the  point  m  (Fig.  1464)  to  m'  (Fig.  1465)  ; 
lay  off  on  the  perpendicular  forming  the  projection,  the  height  m'  o  of  the  cube  ; 
draw  the  lines  m'  D'  and  0D'.  Lay  off  the  distance  m' r  equal  to  mh  (Fig. 
1464),  and  draw  the  line  rp',  its  intersection  with  m'  D'  will  cut  off  m'  h',  equal 
to  m  h  (Fig.  1464),  and  establish  the  angle  h'  of  the  cube.  From  r  lay  off  r  s, 
equal  to  h  g  (Fig.  1464),  draw  sp',  and  its  intersection  with  m  D'  establishes  the 
angle  g'.  From  h'  draw  a  line  vanishing  in  D.  Through  h'  extend  a  line  p  h' 
to  t,  from  t  lay  off  to  the  left  t  a,  equal  to  the  side  of  the  cube  li  e  ;  draw  apf 
and  its  intersection  with  the  line  h'T)  establishes  a  third  point  Y  of  the  cube. 
Upon  these  points  h'  g'  e'  erect  perpendiculars ;  those  upon  h'  and  g'  will,  by 
their  intersection  with  o  D',  determine  h"  g".  Draw  h"  D,  its  intersection 
with  the  perpendicular  at  e'  determine  e".  Draw  g"  D  and  e"  D'  to  their  inter- 
section, and  the  cube  is  complete. 

To  draw  the  Perspective  Projection  of  an  Octagonal  Pillar  in  Angular  Per- 
spective.— Let  A  (Fig.  1466)  be  the  plan  of  the  pillar.  Inclose  it  by  a  square. 
Let  M  M  be  the  base  line,  and  S  the  station  point ;  determine  the  position  of 
the  vanishing  points  for  the  sides  of  the  square  as  in  Fig.  1464,  and  project  the 
square  upon  the  plane  of  the  picture  M  D  D'  M'  by  either  of  the  methods  already 
explained.  These  lines  of  construction  are  omitted,  as  on  the  necessarily  small 
diagrams  they  would  confuse  the  student ;  but  in  drawing  these  examples  to 
the  scale  recommended,  they  might  be  retained.  From  the  angles  of  the  octa- 
gon visible  to  the  spectator  draw  rays  to  the  station  point  S,  project  their  inter- 
section with  the  base  line  M  M,  to  the  perspective  square  (Fig.  1467),  which 
will  thus  determine  on  the  sides  of  the  square  the  positions  of  the  points  a',  b', 
c',  d',  e',  corresponding  to  the  visible  angles  of  the  octagon  ;  connect  these 
points  by  lines.  To  construct  the  pillar  upon  this  base,  let  fall  a  perpendicular 
from  the  corner/  of  the  square  upon  M  M',  at  /set  off  the  height  of  the  pillar  ; 
from  this  point/'  draw  lines  to  the  vanishing  points  D,  D',  and  construct  three 
sides  of  an  upper  square  similar  to  the  lower  one.  The  lines  of  this  square  will 
determine  the  length  of  the  sid,es  of  the  tower,  which  are  the  perpendiculars 
let  fall  upon  a'  br  c'  d'  e'. 

To  construct  a  Circular  Pillar  in  Angular  Perspective. — Let  B  (Fig.  1466) 
be  the  plan  of  the  base  ;  enclose  it  with  a  square  whose  sides  are  parallel  re- 
spectively to  S  M  and  S  M  ;  project  this  square  upon  the  plane  of  the  picture 
(Fig.  1467)  ;  divide  the  plan  into  four  equal  squares  by  lines  parallel  to  the 
sides  ;  draw  rays  through  the  points  h  and  i,  and  project  their  intersection 
with  M  M  upon  the  perspective  square.  From  the  points  h'  and  i'  thus  formed, 
draw  lines  to  vanishing  points  D'  and  D,  and  the  perspective  square  is  divided 


PERSPECTIVE  DRAWING. 


615 


FIG.  1466. 


\  \f 


FIG.  1467. 


£' 


--....r' 


M       m" 


FIG.  1468. 


r 


616  PERSPECTIVE  DRAWING. 

similarly  to  the  original,  and  there  are  four  points  of  the  circle  established  : 
through  these  draw  the  perspective  of  the  circle.  By  the  division  of  the  base 
into  smaller  squares  more  points  of  the  curve  might  be  determined,  but  for  the 
present  purpose  they  are  unnecessary.  To  determine  the  outline  of  the  pillar, 
draw  from  S  rays  tangent  to  the  sides  of  the  plan  at  k  and  i,  the  perpendicu- 
lars let  fall  from  their  intersection  with  M  M  will  be  the  outline  of  the  cylin- 
der. To  cut  them  off  to  the  proper  height,  and  to  determine  the  top  of  the 
cylinder,  upon  the  perpendicular  let  fall  upon  i,  set  off  the  height  of  the  cylin- 
der /'  r,  and  upon  this  plane  project  the  square  as  before,  and  draw  in  through 
the  points  thus  determined  the  outline  of  the  curve.  As  a  still  further  eluci- 
dation of  the  principle  of  projection,  an  enlarged  cap  is  represented  on  the 
pillar,  of  which  the  circumscribing  circle  (Fig.  1466)  is  the  plan.  In  this,  by 
extending  the  central  lines  of  the  square,  both  in  plan  and  perspective,  we  are 
enabled  to  project  readily  eight  points  in  the  larger  circle  through  which  the 
curve  may  be  drawn. . 

To  draw  an  Octagonal  Pyramid  in  Angular  Perspective. — Let  A  (Fig. 
1466)  be  the  base  of  the  pyramid  ;  project  upon  the  plane  of  the  picture  (Fig. 
1468)  the  visible  angles  of  the  base,  as  in  the  case  of  the  pillar.  Through  the 
center  of  the  plan  draw  a  line  parallel  to  one  of  the  sides  and  intersecting  M  M 
at  m  ;  from  this  point  let  fall  a  perpendicular  to  m'  on  M  M'  (Fig.  1468) ;  on 
this  perpendicular  set  off  the  height  of  the  pyramid  m-  o  from  m'  and  draw 
lines  to  D'.  From  the  center  of  the  plan  draw  a  ray  to  S,  and  project  its 
intersection  with  M  M,  upon  the  line  o  D',  its  intersection  o'  with  this  line  will 
be  the  apex  of  the  pyramid  :  from  this  point  draw  lines  to  the  angles  of  the 
base  already  projected,  and  the  pyramid  is  complete. 

To  draw  a  Cone  in  Angular  Perspective. — Let  the  inner  circle  B  (Fig. 
1466)  be  the  base  of  the  cone,  project  its  visible  outline  to  Fig.  1468,  as  in  case 
of  the  cylinder.  To  determine  its  height  extend  one  of  the  diameters  of  the 
plan  to  the  base  line  at  p  ;  from  this  point  let  fall  a  perpendicular  to  p'  on 
M  M',  and  set  off  upon  it  p'  q  the  height  of  the  cone  ;  from  p'  and  q  draw 
lines  to  the  vanishing  point  D'.  From  the  center  of  the  plan  (Fig.  1466)  draw 
a  ray  to  S,  and  project  its  intersection  with  M  M  upon  r'  on  the  line  q  D',  and 
r'  will  be  the  apex  of  the  cone  :  connect  the  apex  with  the  extremities  of  the 
perspective  of  the  base,  and  the  projection  of  the  cone  is  complete. 

To  draw  the  Elevation  of  a  Building  in  Angular  Perspective. — For  ex- 
ample, take  the  school-house  (Fig.  1469).  Plot  so  much  of  the  plan  of  the 
building  as  may  be  seen  from  the  position  of  the  spectator  at  S.  Draw  a 
base  line,  and  through  the  station  point  draw  parallels  to  the  sides  of  the 
building,  cutting  the  base  as  at  M  M  ;  draw  M  M'  for  a  base,  and  D  D'  for  the 
horizontal  line  of  the  picture.  Project  M  and  M  to  D  and  D',  for  the  vanish- 
ing points,  the  one  of  the  lines  parallel  to  a  c,  the  other  to  a  1)  ;  extend  a  c,  ab ; 
project  d,  e,  to  d',  e',  and  on  d'  d  set  off  the  height  of  the  eaves  d'  o,  and  of  the 
ridge  dr  n  ;  from  d',  o  and  n  draw  lines  to  D',  and  from  e'  to  D,  draw  rays  from 
c  and  b  to  S,  and  project  their  intersection  with  the  base  to  the  vanishing  lines 
just  drawn.  To  find  the  perspective  of  the  ridge  draw  a  ray  from  the  center 
of  a  b,  and  project  its  intersection  with  the  base  to  r  on  the  line  nD',  the  point 
is  the  apex  of  the  gable,  the  line  r  D  will  be  the  perspective  of  the  ridge  ;  to 


PERSPECTIVE  DRAWING. 


618  PERSPECTIVE  DRAWING. 

determine  its  length  erect  a  perpendicular  at  the  intersection  of  tD'  and  s  D, 
draw  the  sloping  lines  of  the  roof,  and  the  outline  of  the  building  is  complete. 
The  filling  in  of  the  details  will  be  readily  understood  ;  it  will  only  be  neces- 
sary to  keep  in  mind  that  all  lines  parallel  to  a  b  must  meet  in  D',  those  to  a  c 
in  D  :  all  measures  laid  oif  on  any  lines  of  the  plan  must  be  connected  with 
the  point  of  sight  S,  and  their  intersections  with  the  base  projected.  All  ver- 
tical heights  must  be  laid  oif  on  the  line  d'  d,  and  referred  to  the  proper  posi- 
tion by  lines  to  D  or  D',  as  the  case  may  be. 

As  an  example  of  the  other  method  of  constructing  this  same  problem,  let 
the  scholar  lay  off  to  the  double  of  the  present  scale  the  plane  of  the  picture 
M  D  D'M',  and  the  division  points  p'  and  p,  and  without  drawing  plan  or  ele- 
vation take  the  dimensions  from  Fig.  1190. 

To  draw  an  Arched  Bridge  in  Angular  Perspective. — Let  A  and  B  (Fig. 
1470)  be  the  plans  of  the  piers  ;  on  the  line  a  p,  one  of  the  sides  of  the 
bridge,  lay  down  the  curve  of  the  arch  as  it  would  appear  in  elevation,  in 
this  example  an  ellipse.  Divide  the  width  of  the  arch  as  at  b  c  d  e  f  g  h, 
carry  up  lines  perpendicular  to  b  h  until  they  intersect  the  curve  of  the  arch, 
and  through  these  points  draw  lines  parallel  tobh  as  k  I  m  ;  let  o  r  be  the 
height  of  the  parapet  of  the  bridge  above  the  spring  of  the  arch.  Through 
the  station  point  draw  lines  parallel  to  the  side  a  h  and  end  a  a  of  the  bridge, 
till  they  intersect  the  assumed  base  line  M  M  :  project  these  intersections  to 
the  horizon  line  of  the  picture  for  the  vanishing  points  D,  D'  of  perspective 
lines  parallel  to  a  li  and  a  a.  Let  fall  a  perpendicular  from  a  to  a' ,  and  on  this 
perpendicular  set  off  from  a'  the  heights  s  k,  si,  s  m,  and  s  r  ;  from  a'  and  r' 
draw  lines  to  D  and  D',  and  from  the  points  m',  I',  k1  to  D'.  Draw  rays  from 
the  points  abed efg  h  to  the  station  point  S,  and  project  their  intersection 
with  the  base  lines  to  the  perspective  line  a'  D'  as  in  previous  examples  :  the 
intersection  of  the  lines  k'  D',  I'  D',  m'  D'  by  the  perpendiculars  thus  pro- 
jected will  establish  the  points  of  the  curve  of  the  arch  on  the  side  nearest  the 
spectator.  To  determine  the  position  of  the  opposite  side  of  the  arch,  from  a", 
the  perspective  width  of  the  bridge,  draw  a"  D',  and  from  h'  draw  lines  to  D  ; 
the  line  h'  p'  will  be  the  perspective  width  of  the  pier  ;  draw  k'  D  ;  and  from 
k",  k"  D' ;  from  g"  the  intersection  of  the  curve  of  the  arch  by  the  perpen- 
dicular to  g'9  draw^D,  the  intersection  with  k"D'  will  be  one  point  in  the 
curve  of  the  arch  on  the  opposite  side  of  the  bridge  ;  in  the  same  way,  from 
any  point  in  the  nearer  arc  draw  lines  to  D,  and  the  intersection  with  lines  in 
the  same  planes  on  the  opposite  side  of  the  bridge  will  furnish  points  for  the 
further  arch  ;  all  below  the  first  only  will  be  visible  to  the  spectator. 

To  draw  in  Parallel  Perspective  the  Interior  of  a  Room  (Fig.  1471). — We 
propose  to  construct  this  by  scale  without  laying  down  the  plan.  Draw  the 
horizon  line  D  V  D',  and  the  base  M  M',  making  D  and  D'  the  point  of  dis- 
tance. Let  the  room  be  20  feet  wide,  14  feet  high,  and  12  feet  deep  ;  on  the 
base  M  M'  lay  off  the  rectangle  of  the  section  in  our  figure  on  a  scale  of  8  feet 
to  the  inch,  20  feet  X  14  feet.  From  the  four  corners  draw  lines  to  the  center 
of  view  V  ;  from  S'  lay  off  to  the  right  or  left  on  M  M'  12  feet,  and  through  this 
point  draw  lines  to  D'  or  D  as  the  case  may  be  ;  through  the  point  of  intersec- 
tion, a'  of  this  line  with  S'  V,  draw  a  line  parallel  to  M  M' ;  at  the  intersections 


PERSPECTIVE   DRAWING. 


FIG.  1471. 


620 


PERSPECTIVE  DRAWING. 


of  this  line  with  M  V  and  M'  V  erect  a  perpendicular,  cutting  the  vanishing 
lines  of  the  upper  angle  of  the  room  at  d  and  e  ;  connect  de  and  the  perspect- 
ive of  the  room  is  complete.  To  draw  the  aperture  for  a  door  or  window  on 
the  side,  measure  oil  from  S'  the  distance  of  the  near  side  from  the  plane  of 
the  picture,  and  in  addition  thereto  the  width  of  the  aperture  ;  from  these  two 
points  draw  lines  to  the  proper  point  or  distance,  and  at  their  intersection  with 
S'  V,  draw  parallels  to  MM',  cutting  the  lower  angles  of  the  room,  and  erect 
perpendiculars,  the  height  of  which  will  be  determined  by  a  line  drawn  from/, 
the  height  of  the  window  above  the  floor  measured  011  M  D.  Should  the  win- 
dow be  recessed,  the  farther  jamb  will  be  visible  ;  extend  the  farther  parallel 
to  M  M',  and  cut  it  by  a  line  gV.  M.g  being  the  depth  of  the  recess,  the  rest 
of  the  construction  may  be  easily  understood  by  inspection  of  the  figure.  At 
the  extremity  of  the  apartment  a  door  is  represented  half  open,  hence  as  the 
plane  of  the  door  is  at  right  angles  to  the  plane  of  the  picture,  the  top  and  bot- 
tom lines  will  meet  in  the  point  of  view ;  if  the  door  were  open  at  an  angle  of 
45°  these  lines  would  meet  in  the  points  of  distance  ;  if  at  any  other  angle, 
the  vanishing  points  would  have  to  be  determined  by  constructing  a  plane, 
drawing  a  line  parallel  to  the  side  of  the  door  through  the  station  point,  and 
projecting  it  upon  the  horizon  line.  The  chair  in  the  middle  of  the  room  is 
placed  diagonally,  and  the  table  parallel  to  the  plane  of  the  picture  ;  their  pro- 
jection is  simple. 

To  draw  in  Perspective  a  Flight  of  Stairs  (Fig.  1472). — Lay  off  the  base 
line,  horizon,  center  of  view,  and  point  of  distance  of  the  picture ;  construct 


FIG.  1472. 


the  solid  abed,  efg  h,  containing  the  stairs,  and  in  the  required  position  in  the 
plane  of  the  picture  ;  divide  the  rise  a  c  into  equal  parts  according  to  the  num- 
ber of  stairs,  nine  for  instance  ;  divide  perspectively  the  line  a  b  into  one  less  (8) 


PERSPECTIVE  DRAWING. 


621 


number  of  parts  ;  at  the  points  of  division  of  this  latter  erect  perpendiculars, 
and  through  the  former  draw  lines  to  the  center  of  view  ;  one  will  form  the 
rise  and  the  other  the  tread  of  the  steps.  From  the  top  of  the  first  step  to  the 
top  of  the  upper  continue  a  line  a  d,  till  it  meets  the  perpendicular  S'  V  pro- 
longed in  v  ;  this  line  will  be  the  inclination  or  pitch  of  the  stairs  ;  if  through 
the  top  of  the  step  at  the  other  extremity  a  similar  line  be  drawn,  it  will  meet 
the  central  perpendicular  at  the  same  point  v,  and  will  define  the  length  of  the 
lines  of  nosing  of  the  steps,  and  the  other  lines  may  be  completed.  As  the 
pitch  lines  of  both  sides  of  the  stairs  meet  the  central  vertical  in  the  same 
point,  in  like  manner  v  will  be  the  vanishing  point  of  all  lines  having  a  similar 
inclination  to  the  plane  of  the  picture.  The  projection  of  the  other  flight  of 
stairs  will  be  easily  understood  from  the  lines  of  construction  perpendicular  to 
the  base  line  or  parallel  thereto,  lying  in  planes. 

To  find  the  Reflection  of  Objects  in  the  Water.— Lei  B  (Fig.  1473)  be  a  cube 
suspended  above  the  water  ;  we  find  the  reflection  of  the  point  a,  by  letting 
fall  a  perpendicular  from  it,  and  setting  off  the  distance,  a'  w  below  the  plane 
of  the  water  equal  to  the  line  aw  above  this  line,  the  line  wf  will  also  be 
equal  to  the  line  wf ;  find  in  the  same  way  the  points  V  and  e',  through  these 
points  construct  perspectively  a  cube  in  this  lower  plane,  and  we  have  the  re- 
flection of  the  cube  above. 

To  find  the  reflection  of  the  square  pillar  D  removed  from  the  shore  :  sup- 
pose the  plane  of  the  water  extended  beneath  the  pillar,  and  proceed  as  in  the 
previous  example. 

It  will  be  observed  that  those  lines  of  an  object  which  meet  in  the  center  of 
view  V,  in  the  original,  have  their  corresponding  reflected  lines  converging  to 


D 


B 


FIG.  1473. 

the  same  point.  If  the  originals  converge  to  the  points  of  distance,  the  reflected 
ones  will  do  the  same.  To  find  the  reflection  of  any  inclined  line,  find  the  re- 
flection of  the  rectangle  of  which  it  is  the  diagonal,  if  the  plane  of  the  rectangle 
is  perpendicular  to  the  plane  of  the  picture  ;  if  the  line  is  inclined  in  both 
directions  inclose  it  in  a  parallelepiped  and  project  the  reflection  of  the  solid. 


PERSPECTIVE  DRAWING. 

To  find  the  Perspective  Projection  of  Shadows  (Fig.  1474). — Let  the  con- 
struction points  and  lines  of  the  picture  be  plotted.  Let  A  be  the  perspective 
projection  of  a  cube  placed  against  another  block,  of  which  the  face  is  parallel 
to  the  plane  of  the  picture  ;  to  find  the  shadow  upon  the  block  and  upon  the 
ground  plane,  supposing  the  light  to  come  into  the  picture  from  the  upper 
left-hand  corner  and  at  an  angle  of  45°.  Since  the  angle  of  light  is  the  diagonal 
of  a  cube,  construct  another  cube  similar  to  A,  and  adjacent  to  the  face  dcg  ; 
draw  the  diagonal  b  k,  it  will  be  the  direction  of  the  rays  of  light,  and  k  will 
be  the  shadow  of  b  ;  connect  fk  and  c  k,  fk  must  be  the  shadow  of  the  line 
bf,  and  c  k  of  b  c  ;  the  one  upon  the  horizontal  plane  and  the  other  in  a  verti- 
cal one  :  the  former  will  have  its  direction,  being  a  diagonal,  toward  the  point 
of  distance  D',  the  other  being  a  diagonal  in  a  plane  parallel  to  that  of  the 
picture,  will  be  always  projected  upon  this  plane  in  a  parallel  direction. 

Let  B  be  a  cube  similar  to  A  ;  to  find  its  projection  upon  a  horizontal  plane, 
the  shadow  of  the  point  l>  may  be  determined  as  in  the  preceding  example,  but 
the  shadow  of  the  point  c',  instead  of  falling  upon  a  plane  parallel  to  the  pic- 
ture, falls  upon  a  horizontal  one  ;  its  position  must  be  determined  as  we  did 
before  by  b.  Construct  the  cube  and  draw  the  diagonal  c'  I ;  in  the  same  way 
determine  the  point  m  the  shadow  of  d'  ;  connect  ck'  Im  n,  and  we  have  the 
shadow  of  the  cube  in  perspective  on  a  horizontal  plane. 

On  examination  of  these  projected  shadows,  it  will  be  found  that  as  the 
rays  of  light  fall  in  a  parallel  direction  to  the  diagonal  of  the  cube,  the  vanish- 
ing point  of  these  rays  will  be  in  one  point  V  on  the  line  D'  M'  prolonged,  at 
a  distance  below  D'  equal  V  D'  ;  and  since  the  shadows  of  vertical  lines  upon  a 
horizontal  plane  are  always  directed  toward  the  point  of  sight,  the  extent  of 
the  shadow  of  a  vertical  line  may  be  determined  by  the  intersection  of  the 
shadow  of  the  ground  point  of  the  line  by  the  line  of  light,  from  the  other  ex- 
tremity. Thus,  the  point  k,  cube  A,  is  the  intersection  of  /D'  by  bV  ;  the 
points  k',  I,  ware  the  intersections  of  eD',  oD',  nW  by  V  V,  c'V  d'V. 
Similarly  on  planes  parallel  to  that  of  the  picture,  k,  cube  A  is  intersection  of 
the  diagonal  c  k,  by  the  ray  of  light  b  V. 

Applying  this  rule  to  the  frame  C,  from  r,  s,  p,  draw  lines  to  D' ;  from  rr, 
$',  p'f  draw  rays  to  V  ;  their  intersections  define  the  outline  of  the  shadow  of 
the  post.  To  draw  the  shadow  of  the  projection,  the  shadow  upon  the  post 
from  t  will  follow  the  direction  of  the  diagonal  ck.  Project  u  and  v  upon  the 
ground  plane  at  u'  and  v'  ;  from  t  u'  v'  and  p  draw  lines  to  D'  ;  from  t'9  u,  v, 
w  and  x  draw  rays  to  V,  and  the  intersection  of  these  lines  with  their  cor- 
responding lines  from  their  bases  will  give  the  outline  required  ;  as  v  and  w 
are  on  the  same  perpendicular,  their  rays  will  intersect  the  same  line  v'  V. 

With  reference  to  the  intensity  of  "  shade  and  shadow,"  and  the  necessary 
manipulation  to  produce  the  required  effect,  the  reader  is  referred  to  the  article 
on  this  subject, 

In  treating  of  Perspective  it  has  been  considered  not  in  an  artistic  point,  as 
enabling  a  person  to  draw  from  nature,  but  rather  as  a  useful  art  to  assist  the 
architect  or  engineer  to  complete  his  designs,  by  exhibiting  them  in  a  view 
such  as  they  would  have  to  the  eye  of  a  spectator  when  constructed.  In  our 
examples,  owing  to  size  of  the  page,  we  have  been  limited  in  the  scale  of  the 


PERSPECTIVE  DRAWING. 


623 


624  PERSPECTIVE  DRAWING. 

figures,  and  in  the  distance  of  the  point  of  view,  or  distance  of  the  eye  from, 
the  plane  of  the  picture,  and  as  it  was  unimportant  to  the  mathematical  demon- 
stration, few  of  the  figures  extend  above  the  line  of  the  horizon.  In  these  par- 
ticular points  it  is  unnecessary  that  the  examples  should  be  copied.  The  most 
agreeable  perspective  representations  are  generally  considered  to  be  produced 
by  fixing  the  angle  of  vision  M  S  M',  at  from  45°  to  50°,  and  the  distance  of 
the  horizon  above  the  ground-line  at  about  one  third  the  height  of  the  picture. 
Linear  perspective  is  more  adapted  to  the  representation  of  edifices,  bridges, 
interiors,  etc.,  than  to  that  of  machinery  ;  it  belongs,  therefore,  rather  to  the 
architect  than  to  the  engineer  or  the  mechanic  ;  for  the  purposes  of  the  latter 
we  would  recommend  Isometrical  Perspective,  uniting  accuracy  of  measures 
with  graphic  perspective  representation. 


ISOMETRICAL  DRAWING. 


PKOFESSOR  FARISH,  of  Cambridge,  has  given  the  term  Isometrical  Per- 
spective to  a  particular  projection  which  represents  a  cube,  as  in  Fig.  1474, 
The  words  imply  that  the  measure  of  the  representations  of  the  lines  forming 
the  sides  of  each  face  are  equal. 

The  principle  of  isometric  representation  consists  in  selecting,  for  the  plane 
of  the  projection,  one  equally  inclined  to  three  principal  axes,  at  right  angles 
to  each  other,  so  that  all  straight  lines 

coincident  with  or  parallel  to  these  9_ 

axes  are  drawn  in  projection  to  the 


a 


FIG.  1474. 


same  scale.     The  axes  are  called  iso- 
metric axes,  and  all  lines  parallel  to  FIG.  1475. 
them  are  called  isometric  lines.     The 

planes  containing  the  isometric  axes  are  isometric  planes ;  the  point  in  the 
object  projected,  assumed  as  the  origin  of  the  axes,  is  called  the  regulating- 
point. 

To  draw  the  isometrical  projection  of  a  cube  (Fig.  1475),  draw  the  horizontal 
line  A  B  indefinitely ;  at  the  point  D  erect  the  perpendicular  D  F,  equal  to  one 
side  of  the  cube  required ;  through  D  draw  the  lines  D  b  and  D  /  to  the  right, 
and  left,  making /D  B  and  b  D  A  each  equal  an  angle  of  30°.  Consequently, 
the  angles  F  D  /  and  F  D  b  are  each  equal  to  60°.  Make  D  b  and  D  /  each 
equal  to  the  side  of  the  cube,  and  at  b  and /erect  perpendiculars,  making  b  a 
and/e  each  equal  to  the  side  of  the  cube  ;  connect  F  a  and  F  e,  and  draw  e  g 
parallel  to  a  F,  arid  a  g  parallel  to  F  e,  and  we  obtain  the  projection  of  the 
cube. 

40 


626 


ISOMETRICAL  DRAWING. 


If  from  the  point  F,  with  a  radius  F  D,  a  circle  be  described,  and  commenc- 
ing at  the  point  D  radii  be  laid  oif  around  the  circumference,  forming  a  regular 
inscribed  hexagon,  and  the  points  D  a  e  be  connected  with  the  center  of  the 
circle  F,  we  have  an  isometrical  representation  of  a  cube.  The  point  D  is  called 
the  regulating-point. 

If  a  cube  be  projected  according  to  the  principles  of  isometrical  perspective, 
in  a  similar  manner  as  we  have  constructed  one  according  to  the  rules  of  linear 
perspective,  the  length  of  the  isometrical  lines  would  be  to  the  original  lines  as 
•8164  to  1,  but,  since  the  value  of  isometrical  perspective  as  a  practical  art  lies 
in  the  applicability  of  common  and  known  scales  to  the  isometric  lines,  in  our 
constructions  we  have  not  thought  it  necessary  to  exemplify  the  principles  of 
the  projection,  but  have  drawn  our  figures  without  any  reference  to  what  would 
be  the  comparative  size  of  the  original  and  of  the  projection,  transferring  meas- 
ures directly  from  plans  and  elevations  in  orthographic  projections  to  those  in 
isometry.  It  will  be  observed  that  the  isometric  scale  adopted  applies  only  to 
isometric  lines,  as  F  D,  F  a,  and  F  e,  or  lines  parallel  thereto ;  the  diagonals 
which  are  absolutely  equal  to  each  other,  and  longer  than  the  sides  of  the  cube, 
are  the  one  less,  the  other  greater ;  the  minor  axis  being  unity,  the  isometrical 
lines  and  the  major  axis  are  to  each  other  as,  1.  /y/2.  <\/3. 

Understanding  the  isometrical  projection  of  a  cube,  any  surface  or  solid  may 
be  similarly  constructed,  since  it  is  easy  to  suppose  a  cube  sufficiently  large  to 
contain  within  it  the  whole  of  the  model  intended  to  be  represented,  and,  as 
hereafter  will  be  further  illustrated,  the  position  of  any  point  on  or  wi  fchin  the 
cube,  the  direction  of  any  line,  or  the  inclination  of  any  plane  to  which  it  may 
be  cut,  can  be  easily  ascertained  and  represented. 


FIG.  1476. 


FIG.  1478. 


In  Figs.  1474  and  1475  one  face  of  the  cube  appears  horizontal,  and  the 
other  two  faces  appear  vertical.     If  now  the  figures  bo  inverted,  that  which 


ISOMETRIC  AL  DRAWING. 


627 


before  appeared  to  be  the  top  of  the  object  will  now  appear  to  be  its  under 
side. 

The  angle  of  the  cube  formed  by  the  three  radii  meeting  in  the  center  of 
the  hexagon  may  be  made  to  appear  either  an  internal  or  external  angle  ;  in 
the  one  case  the  faces  representing  the  interior,  and  in  the  other  the  exterior  of 
a  cube. 

Figs.  1476,  1477,  1478,  illustrate  the  application  of  isometrical  drawing  to 
simple  combinations  of  the  cube  and  parallelopipedon.  The  mode  of  construc- 
tion of  these  figures  will  be  easily  understood  by  inspection,  as  they  contain  no 
lines  except  isometrical  ones. 

To  draiv  Angles  to  the  Boundary  Lines  of  an  Isometrical  Cube. — Draw  a 
square  (Fig.  1479)  whose  sides  are  equal  to  those  of  the  isometrical  cube  A 
(Fig.  1480),  and  from  any  of  its  angles  describe  a  quadrant,  which  divide 


4U      JO     20 
FIG.  1479. 


FIG.  1480. 

into  90°,  and  draw  radii  through  the  divisions  meeting  the  sides  of  the 
square.  These  will  then  form  a  scale  to  be  applied  to  the  faces  of  the  cube  ; 
thus,  on  D  E,  or  any  other,  by  making  the  same  divisions  along  their  respec- 
tive edges. 

As  the  figure  is  bounded  by  twelve  isometrical  lines,  and  the  scale  of  tan- 
gents may  be  applied  two  ways  to  each,  it  can  be  applied  therefore  twenty-four 
ways  in  all,  affording  a  simple  means  of  drawing,  on  the  isometrical  faces  of 
the  cube,  lines  at  any  angles  with  their  boundaries. 

Figs.  1481  to  1486  show  the  section  of  the  cube  by  single  planes,  at  various 
inclinations  to  the  faces  of  the  cubes.  Figs.  1487  and  1488  are  the  same  cube, 
but  turned  round,  with  pieces  cut  out  of  it.  Fig.  1489  is  a  cube  cut  by  two 
planes  forming  the  projection  of  a  roof.  Fig.  1490  is  a  cube  with  all  of  the 
angles  cut  off  by  planes,  so  as  to  leave  each  face  an  octagon.  Fig.  1491  repre- 
sents the  angles  cut  off  by  planes  perpendicular  to  the  base  of  the  cube,  form- 
ing thereby  a  regular  octagonal  prism.  By  drawing  lines  from  each  of  the 
angles  of  an  octagonal  base  to  the  center  point  of  the  upper  face  of  the  cube, 
we  have  the  isometrical  representation  of  an  octagonal  pyramid. 

As  the  lines  of  construction  have  all  been  retained  in  these  figures,  they  will 


628 


ISOHETRICAL  DKAWING. 


FIG.  1481. 


FIG.  1482. 


FIG.  1483. 


FIG.  1484. 


FIG.  1485. 


FIG.  1486. 


FIG.  1487. 


FIG.  1488. 


FIG.  1489. 


FIG.  1490 


ISOMETRICAL  DRAWING. 


629 


be  easily  understood  and  copied,  and  are  sufficient  illustrations  of  the  method 
of  representing  any  solid  by  inclosing  it  in  a  cube. 

In  the  application  of  this  species  of  projection  to  curved  lines,  let  A  B  (Fig. 
1492)  be  the  side  of  a  cube  with  a  circle  inscribed  ;  and  that  all  the  faces  of  a 
cube  are  to  have  similarly  inscribed  circles.  Draw  the  diagonals  A  B,  C  D,  and 


FIG.  1492. 


FIG.  1493. 


at  their  intersection  with  the  circumference,  lines  parallel  to  A  C,  B  D.  Now 
draw  the  isometrical  projection  of  the  cube  (Fig.  1493),  and  lay  out  on  the 
several  faces  the  diagonals  and  the  parallels  ;  the  projection  of  the  circle  will 
be  an  ellipse,  of  which  the  diagonals  being  the  axes,  their  extremities  are  de- 
fined by  their  intersections/ 6,  e5,  a  2,  bl,  d3,  c4,  with  the  parallels  ;  having 


thus  the  major  and  minor  axis,  construct  the  ellipse  by  the  trammel,  or,  since 
the  curve  is  tangent  at  the  center  of  the  sides,  we  have  eight  points  in  the 
curve  ;  it  may  be  put  in  by  sweeps  or  by  the  hand. 


630 


ISOMETRICAL  DRAWING. 


To  divide  the  Circumference  of  a  Circle. — First  method  :  On  the  center  of 
the  line  A  B  (Fig.  1494)  erect  a  perpendicular,  C  D,  making  it  equal  to  C  A  or 
C  B  ;  then  from  D,  with  any  radius,  describe  an  arc  and  divide  it  in  the  ratio 
required,  and  draw  through  the  divisions  radii  from  D  meeting  A  B  ;  then 
from  the  isometric  center  of  the  circle  draw  radii  from  the  divisions  on  A  B, 
cutting  the  circumference  in  the  points  required. 

Second  method  :  On  the  major  axis  of  the  ellipse  describe  a  semicircle,  and 
divide  it  in  the  manner  required.  Through  the  points  of  division  draw  lines 
perpendicular  to  A  E,  which  will  divide  the  circumference  of  the  ellipse  in  the 
same  ratio.  On  the  right  hand  of  the  figure  both  methods  are  shown  in  com- 
bination, and  the  intersections  of  the  lines  give  the  .points  in  the  ellipse. 

Fig.  1495  is  an  isometrical  projection  of  a  bevel-wheel,  with  a  half-plan 
(Fig.  1496)  beneath,  and  projected  lines  explanatory  of  the  method  to  be 


FIG.  1496. 

adopted  in  drawing  the  teeth,  and  of  which  only  half  are  shown  as  cut.  It 
will  be  seen,  by  reference  to  the  second  method  given  above  for  the  division  of 
the  circumference  of  a  circle,  that  the  semicircle  is  described  directly  on  the 
major  axis  of  the  ellipse.  In  practice  it  will  be  found  more  convenient,  when 
a  full  drawing  is  to  be  made,  to  draw  the  semicircle  on  a  line  parallel  to  the 
major  axis,  and  entirely  without  the  lines  of  the  main  drawing ;  and  also,  as  in 
the  example  of  the  bevel-gear,  complete  on  the  semicircle,  or  half-plan,  the 


ISOMETKICAL  DRAWING.  631 

drawings  of  all  lines,  the  intersections  of  which  with  circles  it  will  be  necessary 
to  project  on  the  isometrical  drawing. 

Fig.  1497  is  an  isometrical  projection  of  a  complete  pillow-block,  with  its 
hold-down  bolts.  By  reference  to  Fig.  592,  and  Figs.  508  and  509,  it  will 
be  seen  how  much  more  graphically  these  forms  of  gearing  are  given  by  isom- 
etry  than  by  the  usual  projection.  As  an  exercise  for  the  learner,  it  will  be 
very  good  practice  to  project  isometrically  the  spur-gear  (Fig.  583),  and  the 
standard  and  hanger  (Figs.  510  and  515),  of  which  sufficient  details  are  given. 


FIG.  1497 


Fig.  1498  is  an  isometrical  projection  of  a  culvert,  such  as  were  built  be- 
neath the  Croton  Aqueduct,  and  is  a  good  example  of  construction,  and  better 
illustrated  by  the  drawing  than  it  would  be  by  plan  and  elevations. 

Fig.  829  is  an  isometrical  view  of  the  overflow  and  outlet  of  the  Victoria 
and  Regent  Street  sewers  in  the  Thames  embankment. 

Fig.  1499  is  an  isometric  elevation  of  the  roof-truss  (Fig.  896).  No  side- 
view  is  shown  on  the  plate,  but  the  dimensions  of  timber  and  spaces  are  drawn 
as  usual  in  practice. 

Figs.  1500  and  1501  are  the  elevation  and  section  in  isometry  of  the  district 
school-house  given  in  Figs.  1189  and  1190.  To  bring  the  drawing  within  the 
limits  of  the  page,  the  scale  has  been  necessarily  reduced,  but  it  is  given  in 


632 


ISOMETRTOAL  DEAWING. 


ISOMETKICAL  DRAWING. 


633 


the  figure  as  it  should  always  be,  either  drawn  or  written,  on  all  drawings 
to  a  scale,  not  intended  for  mere  pictures  or  illustrations.  The  section  is 
drawn  at  the  height  of  8  feet  above  the  base  course,  and  higher  than  is 


FIG.  1499. 


nsual  in  such  sections,  but  it  was  necessary  on  account  of  the  extra  height 
of  the  window-sill  above  the  floor,  desirable  in  all  school-rooms.  Fig.  1501 
is  more  graphic  than  the  plan  (Fig.  1190),  and,  when  there  are  staircases 
one  above  the  other  in  the  drawing,  they  are  more  intelligibly  expressed  ;  but 
there  is  nothing  in  the  present  drawing  that  can  not  be  nearly  as  well  shown 
by  the  plan,  and  to  a  mechanic,  for  the  purposes  of  construction,  the  plan  is 
the  simpler. 

By  comparing  the  elevation  (Fig.  1500)  with  the  perspective  (Fig.  1469), 
the  former  appears  distorted,  and  out  of  drawing,  but  it  is  much  more  readily 
drawn,  and  has  this  great  convenience,  that  it  is  drawn  to  and  can  be  measured 
by  a  scale,  but  only  on  the  isometric  lines  :  all  others  are  distorted,  too  long  or 
too  short,  as  may  be  seen  in  the  major  and  minor  axes  of  the  bevel-gear  (Fig. 
1496),  or  the  rake-lines  of  the  roof  (Fig.  1499). 

Fig.  1502  is  the  isometrical  projection,  on  the  wave-line  principle,  of  ship 
construction,  from  Russell's  "Naval  Architecture" — as  explained  and  illus- 


634 


ISOMETKICAL  DRAWING. 


FIG.  1501. 


ISOMETRICAL  DRAWING.  635 

trated  on  pages  458  and  459 — and  Fig.  1503,  another  isometrical  drawing  from 
the  same  work. 

We  have  multiplied  examples  of  isometrical  drawing,  to  show  its  applica- 


\\\\\\\\\\ 
\\x\\\  w\ 


bility  to  varied  forms  of  construction,  mechanical,  architectural,  and  naval. 
The  principles  of  this  projection  are  easy  and  intelligible,  and  their  use  should 


636 


ISOMETRIOAL  DRAWING. 


1SOMETRICAL   DRAWING. 


638 


ISOMETRICAL  DRAWING. 


be  extended.  Isometrical  projection  is  especially  valuable  to  the  mechanical 
draughtsman,  explaining  many  constructions  that  could  hardly  be  done  by  any 
amount  of  plans,  elevations,  and  sections,  and  still  uniting  with  pictorial  rep- 
resentation the  applicability  of  a  scale.  For  drawings  for  the  Patent  Office  it 
is  especially  desirable,  in  a  simple  and  practical  form  combining  the  requisites 
of  many  projections  ;  but  as  a  drawing  of  what  could  be  absolutely  seen  by  the 
eye  it  is  not  truthful,  and  therefore,  when  pictorial  illustration  only  is  requisite, 
the  drawing  should  be  in  linear  perspective. 


FIG.  1505. 

In  confirmation  of  the  above,  in  Fig.  1504  is  given  a  drawing  in  perspec- 
tive, in  which  the  point  of  sight  is  above  the  plane  of  the  picture,  and  ap- 
proaching in  general  appearance  to  drawings  in  isometry  ;  and  yet,  having  all 
the  truthfulness  of  sight,  is  much  better  suited  to  the  purpose  for  which  it 
was  intended.  Fig.  1505  is  another  illustration  of  the  same  kind,  in  common 
use  for  business  circulars  and  catalogues. 


FREE-HAND  DRAWING. 

A  DRAUGHTSMAN",  who  has  made  himself  conversant  with  the  rules  of  pro- 
jection as  laid  down  in  this  book,  and  has  applied  these  rules  to  practice,  will 
be  capable  of  representing  correctly  such  objects  as  have  been  illustrated,  or 
make  up  similar  combinations  of  his  own  invention  and  design.  But  natural 
objects,  as  animals,  trees,  rocks,  clouds,  etc.,  can  not  be  imitated  on  paper 
with  the  aid  of  drawing  instruments  ;  outlines  so  varied  can  not  be  copied  in 
this  mechanical  way  ;  it  can  only  be  done  by  what  is  called  free-hand  drawing, 
an  educated  eye  that  can  recognize  proportion  and  position,  and  an  educated 
hand  that  can  execute  and  portray  naturally  things  recognized  by  the  eye,  with 
the  aid  of  pencil,  pen,  crayon,  or  brush.  A  free  hand  adds  largely  to  the  effect 
of  drawings,  where  close  measures  are  not  requisite,  giving  grace  and  beauty  to 
mechanical  designs,  and  is  especially  applicable  to  architectural  ornaments  and 
accessories.  It  will  be  found  impossible  to  draw  many  of  these  in  any  other 
way,  and  there  are  few  drawings  that  do  not  require  some  patching  by  hand — 
short  curves,  which  can  be  thus  done  much  more  readily,  and  connections  of 
lines,  which  can  not  be  done  by  drawing  instruments.  It  has  been  said  before 
that  the  lettering  of  a  plan  or  map  contributes  very  much  to  its  appearance, 
and  as  the  Italian  and  Koman  characters  are  now  almost  universally  used  it  is 
only  by  free  hand  that  they  can  be  made  ornamental  or  graceful. 

The  pencil  or  pen  should  be  held  by  the  thumb  and  first  finger,  and  sup- 
ported and  guided  by  the  second.  The  two  fingers  touching  the  pencil  should 
be  placed  firmly  on  it,  and  be  perfectly  straight,  the  end  of  the  middle  finger 
at  least  one  inch  above  the  point  of  the  pencil.  In  drawing,  it  is  well  to  com- 
mence, as  in  writing,  with  straight  lines.  Lines  vertical,  horizontal,  and  in- 
clined, parallel  to  each  other  and  at  angles,  light  and  strong — short  and  long 
lines,  straight  and  curved,  with  pen,  pencil,  or  crayon  on  paper,  or  chalk  on  a 
board.  Dot  points,  and  draw  lines  between  them,  at  a  single  movement,  with- 
out going  over  them  a  second  time,  and  without  patching.  Besides  direction, 
lines  have  a  definite  length,  and  the  draughtsman  must  practice  himself  in 
drawing  lines  of  equal  lengths,  or  in  certain  proportions  to  each  other. 

Lines  equal  to  each  other  : 

Lines  twice  another  line  : 


Divide  a  line  into  any  number  of  equal  parts  : 
I I  I  I  I 


640 


FREE-HAND  DRAWING. 


The  accuracy  of  these  divisions  may  be  tested  by  a  strip  of  paper  applied 
along  the  line,  marking  off  the  divisions  upon  it,  and  then  slipping  it  along 
one  division,  and  noting  if  the  divisions  on  the  paper  and  line  still  agree.  By 
practice,  the  eye  will  be  able  to  make  these  divisions  almost  accurately.  Having 
acquired  this  skill,  copy  the  triangles  in  the  Geometrical  Problems,  in  their 
proper  proportions,  and  afterwards  squares  and  rectangles. 


FIG.  1506. 


FIG.  1507. 


FIG.  1508. 


Draw  two  lines  (Fig.  1506)  at  right  angles  to  each  other,  and  mark  equal 
distances  on  each  one.  Through  these  points  draw  a  circle  and  a  square. 

Draw  a  circle  and  divide  each  quadrant  into  two  equal  arcs,  and  connect 
the  chords  to  form  an  octagon  (Fig.  1507).  Or,  draw  a  square,  and  cut  off 
the  corners  (Fig.  1508). 

Divide  a  circle  into  six  equal  arcs,  and  connect  the  chords  for  a  hexagon. 

— r 


FIG.  1509. 


FIG.  1510. 


FIG.  1511. 


FIG.  1512. 


Draw  lines  at  right  angles  to  each  other,  with  only  the  opposite  arms  equal, 
and  construct  the  ellipses  (Figs.  1509  and  1510). 
Draw  an  arc  tangent  to  a  straight  line  (Fig.  1511). 


FREE-HAND  DRAWING. 


641 


Draw  two  parallel  lines  (Fig.  1512),  and  connect  them  by  two  equal  and 
reversed  arcs,  tangent  to  each  other,  and  to  the  parallel  lines. 

Draw  a  similar  curve,  with  arcs  perpendicular  to  the  parallels  (Fig.  1513). 
Although  it  will  be  observed  that  in  all  these  problems  guide  or  construction 
lines  are  used,  it  is  not  the  intention  that 
any  use  should  be  made  of  drawing  instru- 
ments,  but  the  construction  should  be 
dependent  entirely  on  eye  and  hand  ;  still 
it  will  be  found,  whether  the  draughts- 
man draws  from  copy  or  nature,  that  it  is 
almost  impossible  to  get  along  well  with- 
out defining  positions  by  some  points  in 
the  pictures,  and  sketching  in  some  defined  FIG.  1513. 

lines  which  may  serve  as  guides.     All  the 

above  examples  are  from  "Geometrical  Problems,"  and  it  will  be  found  good 
practice  to  copy  others. 

Following  this  practice  of  guide  lines,  it  will  be  well  to  copy  the  outlines  of 
architectural  moldings,  of  which  most  of  the  ornaments  are  conventional  rep- 
resentations of  natural  objects. 

In  design,  "  a  true  artistic  end  has  been  accomplished  when  well-observed 
features  of  natural  objects  have  been  chronicled  within  the  conventionalized 
limits  of  a  few  geometric  rules  that  include  proportion,  symmetry,  and  a  proper 
subordination  of  one  part  to  another." 

The  following  example  is  from  the  "Art  Journal"  (trefoil  design): 
"  In  the  equilateral  triangle  (Fig.  1514),  each  side  is  divided  by 
a  dot,  and  from  the  center  of  the  triangle  lines  are  drawn  to  each 
angle,  and  from  the  dot  in  the  middle  of  each  side  to  the  opposite  sides  of  the 
figure.  The  geometrical  plan  of  the  design  is  thus  laid  out,  and  the  figure  is 
easily  filled  in  by  drawing  simple  curves  from  the  center  of  the  form  to  the 

dot  on  each  side  of  it,  and, 
lastly,  filling  in  the  form  of 
the  trefoil  a  little  below  the 
point  of  each  corner  of  the 
triangle. 

"The  square  (Fig.  1515), 
which  is  the  next  form,  is 
developed  in  much  the  same 
manner.  The  sides  are  bi- 
sected, and  from  a  point  in 
the  center  lines  are  carried 

to  each  angle,  and  to  all  the  dots  on  the  sides.  As  in  the  preceding  figure, 
slight  curves  are  made  on  either  of  the  side-lines,  and  the  trefoil  is  added  to 
each  angle,  with  the  base  of  the  middle  leaf  touching  the  transverse  working- 
lines  between  the  sides.  It  will  be  seen  that  the  pentagon  (Fig.  1516)  and  the 
hexagon  (Fig.  1517)  also  are  formed  in  the  same  general  manner,  but  the  pro- 
portion of  the  top  of  the  trefoil  varies  from  its  sides. 

"In  drawing  the  circular  rosette  (Fig.  1518),  the  circumference  should  be 

41 


FIG.  1514. 


FIG.  1515. 


642  FREE-HAND  DRAWING. 

constructed  on  a  vertical  and  a  horizontal  diameter,  with  two  other  diameters 
bisecting  it  at  equal  angles,  which  divide  it  into  eight  sections,  the  half  diame- 
ters, upon  all  of  which  curved  lines  and  the  top  of  the  trefoil  are  made.  A 


FIG.  1516.  FIG.  1517.  FIG.  1518. 

series  of  arcs  may  be  added  at  the  pleasure  of  the  designer.  In  the  two  pieces 
of  molding  (Figs.  1519  and  1520),  the  trefoil  is  inserted  vertically  to  the  sides 
in  one  and  horizontally  in  the  other.  In  the  latter,  a  half  of  the  trefoil  is 
added  upon  the  sides  to  enrich  the  elementary  figure  ;  and  the  double  line  and 


FIG.  1519. 


the  transverse  lines  which  form  the  squares  are  repeated  for  the  sake  of  sym- 
metry, and  as  affording  an  impression  of  agreeable  repose. 

"  It  is  from  such  a  basis  as  this  that  all  these  various  patterns  are  derived, 


FIG.  1520. 


and  they  produce  a  result  which  an  inexperienced  eye,  unaccustomed  to  analyze 
designs,  could  scarcely  resolve  into  its  elements. " 

Figs.  1521-1524  are  other  illustrations  of  the  same  principle,  of  varieties 
of  rosettes  constructed  on  a  similar  plan. 


FREE-HAND  DRAWING. 


643 


All  of  these  designs  can  be  constructed  mechanically,  but  more  grace  is 
given  to  the  design  by  the  filling  in  with  free  hand,  and  it  is  an  excellent  prac- 
tice in  the  execution  of  the  more  elaborate  Saracenic  and  Moorish  diaper  ;  but 


FIG.  1521. 


FIG.  1522. 


FIG.  1524. 


in  all  of  these  where  there  are  repetitions  of  the  same  figures  it  is  usual  to 
draw  but  one,  and  then  transfer  this,  but  the  finish  must  be  in  crayon  or  pencil. 

"Proportions  of  the  Human  Frame."    By  Joseph  Bonomi. 

The  following,  with  the  illustrations,  are  taken  from  the  above  work  : 

"The  human  frame  is  (Figs.  1525  and  1526)  divided  into  four  equal  meas- 
ures, by  very  distinctly  marked  divisions  on  its  structure  and  outward  form  : 

"  1.  From  the  crown  of  the  head  to  a  line  drawn  across  the  nipples. 

"  2.  From  the  nipples  to  the  pubes. 

"  3.  From  the  pubes  to  the  bottom  of  the  patella  (knee-pan). 

"4.  From  the  bottom  of  the  patella  to  the  sole  of  the  foot. 

"Again,  four  measures,  equal  in  themselves,  and  equal  to  those  just  de- 
scribed, and  as  well  marked  in  the  structure  of  the  human  body,  are  seen  when 
the  arms  are  extended  horizontally.  They  are  the  following  : 

"  From  the  tip  of  the  middle  or  longest  finger  to  the  bend  of  the  arm  is 
one  fourth  of  the  height  of  the  person. 

"  From  the  bend  of  the  arm  to  the  pit  of  the  neck  is  another  fourth. 

"  These  two  measures,  taken  together,  make  the  half  of  the  man's  height, 
and  with  those  of  the  opposite  side  equal  the  entire  height. 

"  In  the  figures,  the  differences  in  width  between  the  male  and  female  figures 
are  given  from  the  tables  of  the  Count  de  Clarac  of  the  Apollino  and  the  Venus 
de  Medici.  The  male  figure  is  in  thicker  line  than  the  female,  and  the  measure- 
ments referring  to  it  are  on  your  right  hand,  and  those  referring  to  the  female 
on  your  left. 

"  The  measurements  of  length,  according  to  Vitruvius  and  Leonardo  da 
Vinci,  are  the  same  in  both  sexes,  and  expressed  in  long  horizontal  lines  run- 
ning through  both  the  front  and  profile  figures. 

"Almost  innumerable  are  the  varieties  of  character  to  be  obtained  by  the 
alterations  of  widths,  without  making  any  change  in  the  measurements  of 
length  ;  nevertheless,  some  ancient  statues  differ  slightly  in  these  measure- 
ments of  length. 

"  No  measurement  is  given  in  the  figure  of  the  width  of  the  foot  ;  its  normal 
proportion  should  be  one  sixteenth  of  the  height.  The  views  of  the  foot  (Fig. 
1527)  are  those  of  the  female. 


644 


FRi£E-HAND  DRAWING. 


"  The  scale,  V,  used  is  8  heads  to  the  height ;  parts,  i  of  a  head  ;  and  min- 
utes, TV  of  a  part. 

"  The  whole  height  is  usually  taken  at  8  heads,  but  there  are  slight  differ- 
ences in  the  classic  statues ;  the  height  of  the  Venus  de  Medici  is  equal  to  7 


heads,  3  parts,  10  minutes,  that  of  the  Apollino  of  Florence,  7  heads,  3  parts, 
6  minutes. 

"  When  the  student  is  acquainted  with  the  forms  of  the  body  and  limbs  in 
two  aspects — viz.,  the  front  and  side  views— and  the  normal  proportions  they 
bear  to  each  other,  then  will  follow  the  study  of  the  characteristic  features  of 


FREE-HAND  DRAWING. 


645 


childhood,  youth,  and  mature  age,  and  those  niceties  of  character  that  the 
ancients  invariably  observed  in  the  statues  of  their  divinities,  so  that  in  most 
cases  a  mere  fragment  of  a  statue  could  be  identified  as  belonging  to  this  or 
that  divinity — as,  for  instance,  the  almost  feminine  roundness  of  the  limbs  of 
the  youthful  Bacchus,  the  less  round  and  distinctly  marked  muscles  of  the 
Mercury,  and  of  the  statues  of  the  Athletae. " 

Figure  Drawing. — In  the  album  of  Villard  de  Hennecourt,  which  dates 
from  the  middle  of  the  thirteenth  century,  certain  mechanical  processes  are 
given  to  facilitate  the  composition  and  design  of  figures.  According  to  these 
sketches,  geometry  is  the  generator  of  movements  of  the  human  body,  and  that 
of  animals,  and  serves  to  establish  certain  relative  proportions  of  the  figures. 
From  the  time  of  Villard  sculptors  have  had  these  practical  methods,  which, 
if  they  could  not  inspire  the  artisan  with  genius,  yet  prevented  him  from  fall- 
ing into  gross  faults.  The  pen  sketch  (Fig.  1528)  is  an  example  of  this  prac- 


C       V 


FIG.  1528. 


FIG.  1529. 


tical  process.  In  comparing  this  mode  of  drawing  with  figures  in  the  vignettes 
of  manuscripts,  with  designs  on  glass,  and  even  with  statues  and  bas-reliefs,  we 
must  recognize  the  general  employment  in  the  thirteenth  and  fourteenth  centu- 
ries of  these  geometrical  means,  suited  to  give  figures  not  only  their  propor- 
tions but  also  the  justness  of  their  movement  and  bearing.  Rectifying  the 
canon  of  Villard  in  its  proportions  by  comparison  with  the  best  statues,  nota- 
bly those  in  the  interior  of  the  western  facade  of  the  Cathedral  of  Reims,  we 


646 


FEEE-HAND  DRAWING. 


obtain  the  Fig.  1529.  The  line  A  B,  the  height  of  the  human  figure,  is  divided 
into  seven  equal  parts.  The  upper  division  is  from  the  top  of  the  head  to  the 
shoulders.  Let  C  D  be  the  axis  of  the  figure,  the  line  at  the  breadth  of  the 
shoulders  is  f  of  the  whole  height  A  B.  The  point  E  is  the  center  of  the  line 
0  D  ;  draw  through  this  point  two  lines,  af  and  b  e,  and  from  the  point  g 
two  other  lines,  g  e  and  g  f.  The  line  1)  h  is  the  length  of  the  humerus,  and 
the  line  of  the  knee-pan  is  on  i  k.  The  length  of  the  foot  is  f  of  a  division,  A  1. 
Having  established  these  proportions,  it  will  be  seen  by  the  following  cuts  how 
the  artisan  gave  movements  to  these  figures  when  the  movements  were  not  in 
absolute  profile. 

Suppose  the  weight  of  the  figure  to  be  borne  upon  one  leg  (Fig.  1530),  the 


FIG.  1530. 


FIG.  1531. 


line  ge  becomes  perpendicular,  and  the  axis  op  of  the  figure  is  inclined.  The 
movement  of  the  shoulders  and  trunk  follow  this  inflection  ;  the  axis  of  the 
head  and  the  right  heel  are  in  the  same  vertical  line. 

In  stepping  up  (Fig.  1531)  the  axis  of  the  figure  is  vertical,  and  the  right 
heel  raised  is  on  the  inclined  line  s  t,  while  the  line  of  the  neck  is  on  the  line 
I  m,  and  the  trunk  is  vertical. 

In  Fig.  1532  it  will  be  seen  how  a  figure  can  be  submitted  to  a  violent  move- 
ment and  vet  preserve  the  same  geometrical  trace.  The  figure  is  fallen,  sup- 
ported on  one  knee  and  one  arm,  while  the  other  wards  off  a  blow  ;  the  head 
is  vertical. 


FREE-HAND   DRAWING. 


647 


In  Fig.  1533,  the  left  thigh  being  in  the  line  af,  to  determine  the  position 
of  the  heel  c  on  the  ground,  supposed  to  be  level,  an  arc  is  to  be  described  from 
the  knee-pan  ;  the  line  ef  is  horizontal. 

It  is  clear  that,  in  adopting  these  practical  methods,  all  the  limbs  can  be 
developed  geometrically  without  shortening. 

The  above  is  from  the  "  Dictionnaire  raisonne  de  1' Architecture  "  of  Viollet 
Le  Due,  and  will  supply  to  many  a  ready  means  of  sketching  the  human  figure 


FIG.  1532. 

in  various  attitudes,  naked,  or  in  the  close-fitting  dresses  of  the  present  fashion  ; 
but  in  the  arrangement  of  drapery  upon  a  figure,  care  must  be  taken  that  the 
drapery  should  fall  in  graceful  folds.  "  It  is  necessary  to  give  the  body  certain 
inflections  which  would  be  ridiculous  in  a  person  walking  naked.  The  walk 
should  be  from  the  hips,  with  wide-spread  legs,  and,  by  the  movements  of  the 
trunk,  make  the  drapery  cling  on  certain  parts  and  float  on  others." 


FIG.  1533. 


In  figures  in  repose,  their  centers  of  gravity  must  fall  within  the  points  of 
support,  but  the  body  can  be  sustained  by  muscular  exertion,  and  this  should 


648 


FREE-HAND  DRAWING. 


FIG.  1535. 


FIG.  1536. 


FIG.  1540. 


•  FIG.  1539. 


FIG.  1541. 


FREE-HAND  DRAWING. 


649 


FIG.  1542. 


FIG.  1546. 


FIG.  1549. 


FIG.  1544. 


FIG.  1545c 


FIG.  1551. 


FIG.  1552. 


650 


FREE-HAND  DRAWING. 


be  expressed  in  such  cases  by  the  tension  of  the  muscles  on  which  the  position 
depends.  In  the  act  of  running,  the  body  inclines  forward,  its  weight  assists 
the  movement,  and  the  motions  prevent  its  falling. 

Figs.  1534-1538  are  illustrations  of  portions  of  the  human  head  and  face, 
with  some  guide-lines  to  assist  the  copyist. 

Figs.  1539-1541  are  drawings  of  female  hands  and  arms. 

Figs.  1542-1545  are  drawings  of  male  hands,  Figs.  1546-1552  of  legs  and 
feet,  with  guide-lines,  and  Figs.  1553-1556  are  those  of  children. 


FIG.  1553. 


FIG.  1554. 


FIG.  1555. 


FIG.  1556. 


The  Forms  of  Animals. — The  bodies  of  most  quadrupeds  standing  can  be  in- 
cluded in  rectangles  as  guide-lines  ;  that  of  the  ox  and  horse  in  that  of  a  square 
(Figs.  1557  and  1558).  The  action  of  the  limbs  of  quadrupeds  is  chiefly  di- 
rectly forward  or  directly  backward,  the  power  of  lateral  motion  being  limited. 
The  hinder  limbs  always  commence  progressive  motion,  as  in  the  first  position 


FREE-HAND   DRAWING. 


651 


of  the  walk  (Fig.  1559),  the  fore  foot  of  the  same  side  advances  next,  then  the 
hind  foot  of  the  opposite  side,  and  lastly  the  fore  foot  on  that  side,  and  so  on. 
In  the  trot,  the  hinder  leg  of  one  side  and  the  fore  leg  of  the  other  are  raised 
together  (Fig.  1560).  In  the  canter  or  gallop,  both  fore  legs  and  one  hind 


FIG.  1557. 


leg  are  raised  together  (Fig.  1561)  ;  when  rapidly  moving,  the  two  fore  legs, 
and  two  hind  legs  appear  to  advance  together  (Fig.  1562).  In  fact,  all  the 
movements  are  rather  resultants,  as  they  appear  to  us,  but  when  instantaneous- 
ly photographed  the  legs  are  wonderfully  mixed. 


FIG.  1558. 


The  forms  of  feet  range  under  two  great  divisions — hoofs  (Fig.  1564)  and 
paws  (Fig.  1565).  All  hoofs,  whether  whole  or  cloven,  approximate  to  a  right- 
angled  triangle,  and  all  paws  to  a  rhomboid. 


652 


FREE-HAND  DRAWING. 


12&*«>r  ~| ^ 


FIG.  1563. 


FREE-HAND  DRAWING. 


653 


the  horse  ;  Fig.  1567, 
ivori ;  Fig.  1569,. 


A 

FIG.  1564. 


ZZ7 


TJie  Noses  of  Animals. — Fig.  1566  represents 
fchat  of  the  ox  and  deer  tribe  ;  Fig.  1568,  those  of 
those   of    the    camel,   sheep,  and 
goat  tribes  ;  and  Fig.  1570,  those 
of  the  hog  tribes.     The  muzzles 
of  nearly  all  quadrupeds  will  be 
found  to  range  under  one  or  other 
of  these  classes,  with  minute  varia- 
tions to  characterize  the  diiferent 
species  and  individuals. 

In  looking  over  the  varied 
sketches  and  engravings  of  Land- 
seer  which  have  been  published,  it 
will  be  noticed  in  how  varied  a 
manner  they  are  executed.  Some- 
times in  mere  outline  with  lead- 
pencil,  sometimes  with  a  camel's- 
hair  pencil  charged  with  Indian 
ink  or  sepia  for  the  outlines,  giv- 
ing effect  to  the  subject  by  slight 
tints  or  washes  of  the  same  color  ; 
in  others,  pen  and  ink  have  been 
alone  employed.  Some  are  in  oils, 
others  in  water-colors  ;  frequent- 
ly chalks,  both  black  and  colored, 
were  the  vehicles  used.  "  As 
we  look  at  some  of  these,  we  are 
tempted  to  believe  that,  of  all  the 
instruments  that  can  be  used  by 
the  artist,  there  is  none  quite  so 
wonderful  as  the  pen.  A  simple 
sketch  with  a  pen  or  lead-pencil  is 
naked,  unadorned  truth,  bearing 
witness  to  the  skill  or  its  opposite 
of  the  hand  which  produced  it." 

The  above  quotation  is  given 
to  show  the  value  of  accurate 
drawing — the  skeleton,  as  it  were, 
may  be  more  suggestive,  and  con- 
vey more  skillfully  effective  truth 
than  the  finished  drawing,  and 
the  first  necessity  is  truth  in  draw- 
ing. Nothing  has  yet  been  said 
of  drawing  from  nature.  The 

copies  given  are  intended  as  rudiments,  and  the  following  illustrations  from 
the  "  Art  Journal"  of  objects  in  art,  and  sketches  and  pictures  of  different 
painters,  will  serve  to  show  their  varied  treatment  of  subjects. 


FIG.  1570. 


654 


FKEE-HAND  DRAWING. 


The  illustrations  given  are  for  the  education  of  the  eye  of  the  draughtsman, 
in  showing  him  the  varied  appearance  of  different  subjects  by  different  artists, 
and  their  modes  of  expression ;  and  he  can  acquire  facility  of  hand  in  copying 
them.  If  he  wishes  to  draw  from  nature,  let  him  look  at  objects  as  if  they 
were  a  picture,  If  he  looks  through  a  window,  the  frame  may  be  considered 
the  border  of  his  picture  ;  if  he  can  portray  what  he  sees  through  a  square  of 
glass  truthfully,  in  position  and  proportion,  with  pencil,  chalk,  or  brush,  he 
has  made  a  picture.  He  must  keep  his  eyes  in  one  position,  or  at  such  a  dis- 
tance from  the  plane  of  his  picture  or  the  glass  that  he  can  not  see  more  of  an 
object  than  is  comprehended  by  one  look.  To  enable  one  to  judge  of  the  pro- 
portion of  an  object,  and  its  position,  it  is  very  common  to  make  use  of  the 
pencil  as  a  scale,  holding  it  with  an  extended  arm  always  at  the  same  distance 
from  the  eye ;  to  slide  the  thumb  down  on  the  pencil  till  the  length  of  the 
object  or  line  is  embraced  between  the  end  of  the  pencil  and  the  thumb,  and 
transferring  this  length  to  the  paper  in  its  proper  position.  Practically,  in 
this  way,  one  arrives  at  the  knowledge  of  perspective,  of  which  the  principles  have 
been  given  in  "  Perspective  Drawing."  Aerial  perspective,  or  the  tones  of  lights 
and  shadows  according  to  their  distances  from  the  observer  and  the  sources  of 
the  light,  he  will  acquire  by  studies  of  pictures  and  observations  of  nature. 
The  rule  in  drawing  from  nature  is  to  draw  only  what  you  see,  and  express  it 
in  the  most  truthful  form. 


FREE-HAND   DRAWING. 


655 


656 


FREE-HAND  DRAWING. 


FREE-HAND    DRAWING. 


657 


Bacchus  and  the  Water- Thieves.     JOHN  PENNIEL. 


ess 


FREE-HAND   DRAWING. 


After  a  Pen-and-ink  Design,  by  FORTUNY. 


FREE-HAND   DRAWING. 


659 


660 


FREE-HANI)   DRAWING. 


I 


Study  of  Oak-Trees.      K.   LAXDSEER. 


FREE-HAND  /DRAWING. 


,^KSi!^      :  ,  «:  ,.^? 


*  %^ 


Apple- -Blossom*.    A.  T.  BRICHER. 


662 


FREE-HAND   DRAWING. 


Cattle  going  Home.    JAMES  M.  HART. 


FREE-HAND   DRAWING. 


663 


Morning.    H.  W.  BOBBINS. 


664 


FREE-HAND    DRAWING. 


m 
I 


,  I 


I 


^ 


1 


APPENDIX. 


Extracts  from  the  Acts  relating  to  Buildings  in  the  City  of  New  Yor%. 

§  3.  All  foundation  walls  shall  be  laid  not  less  than  4'  below  the  surface  of  the  earth, 
on  a  good  solid  bottom,  and,  in  case  the  nature  of  the  earth  should  require  it,  a  bottom  of 
driven  piles,  or  laid  timbers,  of  sufficient  size  and  thickness,  shall  be  laid  to  prevent  the 
walls  from  settling,  the  top  of  such  pile  or  timber  bottom  to  be  driven  or  laid  below  the 
water  line  ;  and  all  piers,  columns,  posts,  or  pillars  resting  on  the  earth,  shall  be  set  upon 
a  bottom  in  the  same  manner  as  the  foundation  walls.  Whenever  in  any  case  the  founda- 
tion wall  or  walls  of- any  building  that  may  hereafter  be  erected  shall  be  placed  on  a  rock 
bottom,  the  said  rock  shall  be  graded  off  level  to  receive  the  same.  .  .  . 

§  4.  The  footing,  or  base  course,  under  all  foundation  walls,  and  under  all  piers,  col- 
umns, posts,  or  pillars  resting  on  the  earth,  shall  be  of  stone  or  concrete ;  and  if  under  a 
foundation  wall  shall  be  at  least  12"  wider  than  the  bottom  width  of  the  said  wall ;  and  if 
Binder  piers,  columns,  posts,  or  pillars,  shall  be  at  least  12"  wider  on  all  sides  than  the 
bottom  width  of  the  said  piers,  columns,  posts,  or  pillars,  and  not  less  than  18"  in  thick- 
ness ;  and  if  built  of  stone,  the  stones  thereof  shall  not  be  less  than  2'  x  3',  and  at  least  8" 
in  thickness ;  and  all  base  stones  shall  be  well  bedded  and  laid  edge  to  edge ;  and  if  the 
walls  be  built  of  isolated  piers,  then  there  must  be  inverted  arches,  at  least  12"  thick, 
turned  under  and  between  the  piers,  or  two  footing  courses  of  large  stone  at  least  10" 
thick  in  each  course.  All  foundation  walls  shall  be  built  of  stone  or  brick,  and  shall  be 
laid  in  cement  mortar,  and,  if  constructed  of  stone,  shall  be  at  least  8"  thicker  than  the 
wall  next  above  them,  to  a  depth  of  16'  below  the  curb  level,  and  shall  be  increased  4"  in 
thickness  for  every  additional  5'  in  depth  below  the  said  16';  and  if  built  of  brick,  shall 
be  at  least  4"  thicker  than  the  wall  next  above  them  to  a  depth  of  16'  below  the  curb  level, 
-and  shall  be  increased  4"  in  thickness  for  every  additional  5'  in  depth  below  the  said  16'. 

§  5.  In  all  dwelling-houses  that  may  hereafter  be  erected  not  more  than  55'  in  height, 
the  walls  shall  not  be  less  than  12"  thick,  and  if  above  55'  in  height,  and  not  more  than 
SO'  in  height,  the  outside  walls  shall  not  be  less  than  16"  thick  to  the  top  of  second  story 
floor-beams  ;  provided  the  same  is  20'  above  the  curb  level,  and  if  not,  then  to  under  side 
of  the  third  story  beams,  and  also  provided  that  portion  of  the  wall, that  is  12"  thick  shall  not 
exceed  40'  above  the  said  16"  wall;  and  in  every  dwelling-house  hereafter  erected  more 
than  80'  in  height,  4"  shall  be  added  to  the  thickness  of  the  wall  for  every  15'  or  part  thereof 
that  is  added  to  the  height  of  the  building.  All  party-walls  in  dwellings  over  55' in  height 
shall  not  be  less  than  16"  in  thickness. 

§  6.  In  all  buildings  other  than  dwellings  hereafter  erected,  the  bearing  walls  shall  not 
be  less  than  12"  thick  to  the  height  of  40'  above  the  curb  level ;  if«  above  40'  in  height 
and  not  more  than  55'  feet  in  height,  the  bearing  walls  shall  not  be  less  than  16"  thick ;  if 
above  55'  and  not  more  than  70'  in  height,  the  bearing  walls  shall  not  be  less  than  20" 


660  APPENDIX. 

thick,  to  the  height  of  20'  above  the  curb  level  or  to  the  next  tier  of  floor-beams  above, 
and  not  less  than  16"  from  thence  to  the  height  of  55'  above  the  curb  level  or  to  the  next 
tier  of  floor-beams,  and  not  less  than  12"  thick  from  thence  to  the  top  ;  and  if  above  TO' 
and  not  more  than  85'  in  height,  the  bearing  walls  shall  not  be  less  than  24"  thick  to  the 
height  of  12'  above  the  curb  level  or  the  second  story  floor-beams,  and  from  thence  to  the 
height  of  60'  above  the  curb  level,  the  said  walls  shall  not  be  less  than  20"  thick,  and  from 
thence  to  the  top  not  less  than  16''  thick  ;  and  if  above  the  height  of  85',  the  bearing  walls 
shall  be  increased  4"  in  thickness  for  every  15',  or  part  thereof,  that  shall  be  added  to  the 
height  of  said  wall  above  the  85'.  In  all  buildings  over  25'  in  width,  and  not  having  either 
brick  partition  walls  or  girders  supported  by  columns  running  from  front  to  rear,  the  wall 
shall  be  increased  an  additional  4"  in  thickness,  to  the  same  relative  thickness  in  height  as 
required  under  this  section  for  every  additional  10'  in  width  of  said  building,  or  any  por- 
tion thereof.  It  is  understood  that  the  amount  of  materials  specified  may  be  used  either 
in  piers  or  buttresses,  provided  the  outside  walls  between  the  same  shall  in  no  case  be  less 
than  12"  in  thickness  to  the  height  of  40',  and  if  over  that  height  then  16"  thick  ;  but  in 
no  case  shall  a  party  wall  between  the  piers  or  buttresses  of  a  building  be  less  than  16"  in 
thickness.  In  all  buildings  hereafter  erected,  situated  on  the  street  corner,  the  bearing 
wall  thereof  (that  is,  the  wall  on  the  street  upon  which  the  beams  rest)  shall  be  4"  thicker 
in  all  cases  than  is  otherwise  provided  for  by  this  act.  All  walls  other  than  bearing  walls 
may  be  4"  less  in  thickness  than  required  in  the  clauses  and  provisions  of  this  section  above 
set  forth,  provided  no  wall  is  less  than  12"  in  thickness. 

§  7.  Every  building  hereafter  erected  more  than  30'  in  width,  except  churches,  thea- 
tres, school-houses,  car-stables,  and  other  public  buildings,  shall  have  one  or  more  stone  or 
brick  partition  walls  running  from  front  to  rear,  or  iron  or  wooden  girders  supported  on 
iron  or  wooden  columns ;  these  walls  shall  be  so  located  that  the  space  between  any  two 
of  the  bearing  walls  shall  not  be  over  25'.  In  case  iron  or  wooden  girders,  supported  on 
iron  or  wooden  columns,  are  substituted  in  place  of  the  partition  walls,  the  building  may 
be  75'  in  width,  but  not  more ;  and  if  there  should  be  substituted  iron  or  wooden  girders, 
supported  on  iron  or  wooden  columns,  in  place  of  partition  walls,  they  shall  be  made  of 
sufficient  strength  to  bear  safely  the  weight  of  250  Ibs.  for  every  square  foot  of  the  floor 
or  floors  that  rest  upon  them,  exclusive  of  the  weight  of  material  employed  in  their  con- 
struction, and  shall  have  a  footing  course  and  foundation  wall  not  less  than  16"  in  thick- 
ness, with  inverted  arches  under  and  between  the  columns,  or  two  footing  courses  of  large, 
well-shaped  stone,  laid  crosswise,  edge  to  edge,  and  at  least  10"  thick  in  each  course,  the 
lower  footing  course  to  be  not  less  than  2'  greater  in  area  than  the  size  of  the  column  ; 
and  under  every  column,  as  above  set  forth,  a  cap  of  cut  granite,  at  least  12"  thick,  and  of 
a  diameter  12"  greater  each  way  than  that  of  the  column,  and  must  be  laid  solid  and  level 
to  receive  the  column.  Any  building  that  may  hereafter  be  erected  in  an  isolated  position, 
and  more  than  100'  in  depth,  and  which  shall  not  be  provided  with  cross  walls,  shall  be 
securely  braced,  both  inside  and  out,  during  the  whole  time  of  its  erection,  if  it  can  be 
done ;  but  in  case  the  same  can  not  be  so  braced  from  the  outside,  then  it  shall  be  properly 
braced  from  the  inside,  and  the  braces  shall  be  continued  from  the  foundation  upward  to 
at  least  one  third  the  height  of  the  building  from  the  curb  level. 

§  8.  ...  Every  temporary  support  placed  under  any  structure,  wall,  girder,  or  beam 
during  the  erection,  finishing,  alteration,  or  repairing  of  any  building,  or  part  thereof,  shall 
be  equal  in  strength  to  the  permanent  support  required  for  such  structure,  wall,  girder,  or 
beam.  And  the  walls  of  every  building  shall  be  strongly  braced  from  the  beams  of  each 
story  until  the  building  is  topped  out,  and  the  roof  tier  of  beams  shall  be  strongly  braced 
to  the  beams  of  the  story  below  until  all  the  floors  in  the  said  building  are  laid. 

§  9.  All  stone  walls  less  than  24"  thick  shall  have  at  least  one  header,  extending 
through  the  walls,  in  every  3'  in  height  from  the  bottom  of  the  wall,  and  in  every  4'  in 
length ;  and,  if  over  24"  in  thickness,  shall  have  one  header  for  every  six  superficial  feet 


APPENDIX.  66T 

on  both  sides  of  the  wall,  and  running  into  the  wall  at  least  2';  all  headers  shall  be  at 
least  18"  in  width  and  8"  in  thickness,  and  shall  consist  of  a  good  flat  stone,  dressed  on  all 
sides.  In  every  brick  wall  every  sixth  course  of  brick  shall  be  a  heading  course,  except 
where  walls  are  faced  with  brick,  in  which  case  every  fifth  course  shall  be  bonded  into 
the  backing  by  cutting  the  course  of  the  faced  brick,  and  putting  in  diagonal  headers 
behind  the  same,  or  by  splitting  face-brick  in  half,  and  backing  the  same  by  a  continuous 
row  of  headers.  In  all  walls  which  are  faced  with  thin  ashlar,  anchored  TO  the  backing, 
or  in  which  the  ashlar  has  not  either  alternate  headers  and  stretchers  in  each  course,  or 
alternate  heading  and  stretching  courses,  the  backing  of  brick  shall  not  be  less  than  12" 
thick,  and  all  12"  backing  shall  be  laid  up  in  cement  mortar,  and  shall  not  be  built  to  a 
greater  height  than  prescribed  for  12"  walls.  All  heading  courses  shall  be  good,  hard, 
perfect  brick.  The  backing  in  all  walls,  of  whatever  material  it  may  be  composed,  shall, 
be  of  such  thickness  as  to  make  the  walls,  independent  of  the  facing,  conform  as  to  thick- 
ness with  the  requirements  of  sections  five  and  six  of  this  act. 

§  10.  Every  isolated  pier  less  than  ten  superficial  feet  at  the  base,  and  all  piers  sup- 
porting a  wall  built  of  rubble-stone  or  brick,  or  under  any  iron  beam  or  arch  girder,  or 
arch  on  which  a  wall  rests,  or  lintel  supporting  a  wall,  shall,  at  intervals  of  not  less  than 
30"  in  height,  have  built  into  it  a  bond  stone  not  less  than  4"  thick,  of  a  diameter  each 
way  equal  to  the  diameter  of  the  pier,  except  that  in  piers  on  the  street  front,  above  the 
curb,  the  bond  stone  may  be  4"  less  than  the  pier  in  diameter  ;  and  all  piers  shall  be  built 
of  good,  hard,  well-burned  bricks  and  laid  in  cement  mortar,  and  all  bricks  used  in  piers 
shall  be  of  the  hardest  quality,  and  be  well  wet  when  laid ;  and  the  walls  and  piers  under 
all  compound,  cast-iron,  or  wooden  girders,  iron  or  other  columns,  shall  have  a  bond  stone 
at  least  4"  in  thickness,  and  if  in  a  wall  at  least  2'  in  length,  running  through  the  wall, 
and  if  in  a  pier,  the  full  size  of  the  thickness  thereof,  every  30"  in  height  from  the  bot- 
tom, whether  said  pier  is  in  the  wall  or  not,  and  shall  have  a  cap  stone  of  cut  granite,  at 
least  12"  in  thickness,  by  the  whole  size  of  the  pier,  if  in  a  pier,  and  if  in  a  wall  it  shall  be 
at  least  2'  in  length,  by  the  thickness  of  the  wall,  and  at  least  12"  in  thickness.  In  any 
case  where  any  iron  or  other  column  rests  on  any  wall  or  pier  built  entirely  of  stone  or 
brick,  the  said  column  shall  be  set  on  a  base  stone  of  cut  granite,  not  less  than  8"  in  thick- 
ness by  the  full  size  of  the  bearing  of  the  pier,  if  on  a  pier,  and  if  on  a  wall  the  full  thick- 
ness of  the  wall.  In  all  buildings  where  the  walls  are  built  hollow,  the  same  amount  of 
stone  or  brick  shall  be  used  in  their  construction  as  if  they  were  solid,  as  above  set  forth ; 
and  no  hollow  walls  shall  be  built  unless  the  t\vo  walls  forming  the  same  shall  be  con- 
nected by  continuous  vertical  ties  of  the  same  materials  as  the  walls,  and  not  over  24"" 
apart.  The  height  of  all  walls  shall  be  computed  from  the  curb  level.  No  swelled  or- 
refuse  brick  shall  be  allowed  in  any  wall  or  pier ;  and  all  brick  used  in  the  construction, 
alteration,  or  repair  of  any  building,  or  part  thereof,  shall  be  good,  hard,  well-burned 
brick;  and  if  used  during  the  months  from  April  to  November,  inclusive,  shall  be  well 
wet  at  the  time  they  are  laid. 

§  12.  In  no  case  shall  the  side,  end,  or  party  wall  of  any  building  be  carried  up  more 
than  two  stories  in  advance  of  the  front  and  rear  walls.  The  front,  rear,  side,  end,  and 
party  walls  of  any  building  hereafter  to  be  erected  shall  be  anchored  to  each  other  every 
6'  in  their  height  by  tie-anchors,  made  of  one  and  a  quarter  inch  by  three  eighths  of  an  inch 
of  wrought-iron.  The  said  anchors  shall  be  built  into  the  side  or  party  walls  not  less  than 
16",  and  into  the  front  and  rear  walls  at  least  one  half  the  thickness  of  the  front  and  rear 
walls,  so  as  to  secure  the  front  and  rear  walls  to  the  side,  end,  or  party  walls ;  and  all 
stone  used  for  the  facing  of  any  building,  except  where  built  with  alternate  headers  and 
stretchers,  as  hereinbefore  set  forth,  shall  be  strongly  anchored  with  iron  anchors  in  each 
stone,  and  all  such  anchors  shall  be  let  into  the  stone  at  least  1".  The  side,  end,  or  party 
walls  shall  be  anchored  at  each  tier  of  beams,  at  intervals  of  not  more  than  eight  feet  apart,, 
with  good,  strong,  wrought-iron  anchors,  one  half  inch  by  one  inch,  well  built  into  the 


£68  APPENDIX. 

-side  walls,  and  well  fastened  to  the  side  of  the  beams  by  two  nails,  made  of  wrought-iron, 
at  least  one  fourth  of  an  inch  in  diameter ;  and  where  the  beams  are  supported  by  girders, 
the  ends  of  the  beams  resting  on  the  girder  shall  be  butted  together  end  to  end,  and 
strapped  by  vvrought-iron  straps  of  the  same  size,  and  at  the  same  distance  apart,  and  in 
the  same  beam  as  the  wall-anchors,  and  shall  be  well  fastened. 

§  13.  All  walls  of  any  buildings  over  fifteen  feet  high  shall  be  built  up  and  extended  at 
least  24"  above  the  roof,  and  shall  be  coped  with  stone  or  iron.  .  .  . 

§  14.  All  iron  beams  or  girders  used  to  span  openings  over  6'  in  width,  and  not  more 
than  12'  in  width,  upon  which  a  wall  rests,  shall  have  a  bearing  of  at  least  12"  at  each 
end  by  the  thickness  of  the  wall  to  be  supported ;  and  for  every  additional  foot  of  span 
over  and  above  the  said  12',  if  the  supports  are  iron  or  solid  cut  stone,  the  bearing  shall 
be  increased  half  an  inch  at  each  end  ;  but  if  supported  on  the  ends  by  walls  or  piers  built 
•of  brick  or  stone,  if  the  opening  is  over  12'  and  not  more  than  18',  the  bearing  shall 
be  increased  4"  at  each  end  by  the  thickness  of  the  wall  to  be  supported  ;  and  if  the  space 
is  over  18'  and  not  more  than  25'  then  the  bearing  shall  be  at  least  20"  at  each  end  by  the 
thickness  of  the  wall  to  be  supported  ;  and  for  every  additional  5'  or  part  thereof  that  the 
space  shall  be  increased,  the  bearing  shall  be  increased  an  additional  4"  at  each  end  by  the 
thickness  of  the  wall  to  be  supported.  And  on  the  front  of  any  building  where  the  sup- 
ports are  of  iron  or  solid  cut  stone,  they  shall  be  at  least  16"  on  the  face  and  the  width  of  the 
thickness  of  the  wall  to  be  supported,  and  shall,  when  supported  at  the  ends  by  brick  walls 
or  piers,  rest  upon  a  cut  granite  base  block,  at  least  12"  thick  by  the  full  size  of  the  bear- 
ing; and  in  case  the  opening  is  less  than  12',  the  granite  block  may  be  6"  in  thickness  by 
-the  whole  size  of  the  bearing ;  and  all  iron  beams  or  girders  used  in  any  buildings  shall 
be,  throughout,  of  a  thickness  not  less  than  the  thickness  of  the  wall  to  be  supported.  All 
iron  beams  or  girders  used  to  span  openings  more  than  8'  in  width,  and  upon  which  a  wall 
rests,  shall  have  wrought  iron  tie-rods  of  sufficient  strength,  well  fastened  at  each  end  of 
the  beam  or  girder,  and  shall  have  cast-iron  shoes  on  the  upper  side,  to  answer  for  the 
skew-back  of  a  brick  or  cut-stone  arch,  which  said  arch  shall  always  be  turned  over  the 
same,  and  the  arch  shall  in  no  case  be  less  than  12"  in  height  by  the  width  of  the  wall  to 
be  supported,  and  the  shoes  shall  be  made  strong  enough  to  resist  the  pressure  of  the  arch 
in  all  cases.  Cut-stone  or  hard-brick  arches,  with  two  wrought-iron  tie-rods  of  sufficient 
strength,  may  be  turned  over  any  opening  less  than  30',  provided  they  have  skew-backs  of 
cut  stone  or  cast  or  wrought-iron,  with  which  the  bars  or  tension-rods  shall  be  properly 
secured  by  heavy  wrought  iron  washers,  necks,  and  heads  of  wrought-iron,  properly 
secured  to  the  skew-backs.  The  above  clause  is  intended  to  meet  cases  where  the  arch 
has  not  abutments  of  sufficient  size  to  resist  its  thrust.  All  lintels  hereafter  placed  over 
openings  in  the  front,  rear,  or  side  of  a  building,  or  returned  over  a  corner  opening,  when 
supported  by  brick  piers  or  iron  or  stone  columns,  shall  be  of  iron,  and  of  the  full  breadth 
of  the  wall  to  be  supported,  and  shall  have  a  brick  arch  of  sufficient  thickness,  with  skew- 
backs  and  tie-rods  of  sufficient  strength  to  support  the  superincumbent  lateral  weight, 
independent  of  the  cast-iron  lintel.  .  .  . 

§  15.  All  openings  for  doors  and  windows  in  all  buildings,  except  as  otherwise  pro- 
vided, shall  have  a  good  and  sufficient  arch  of  stone  or  brick,  well  built  and  keyed,  and 
with  good  and  sufficient  abutments,  or  a  lintel  of  stone  or  iron,  as  follows :  .  .  .  For  an 
opening  exceeding  6'  in  width,  and  not  more  than  8'  in  width,  the  lintel  shall  be  of  iron  or 
stone,  and  of  the  full  thickness  of  the  wall  to  be  supported  :  and  every  such  opening  6'  or 
less  in  width  in  all  walls  shall  be  at  least  one  third  the  thickness  of  the  walls  on  which  it 
rests,  and  shall  have  a  bearing  at  each  end  not  less  than  4"  on  the  walls ;  and  on  the  inside 
of  all  openings,  in  which  the  lintel  shall  be  less  than  the  thickness  of  the  wall  to  be  sup- 
ported, there  shall  be  a  good  timber  lintel  on  the  inside  of  the  other  lintels,  which  shall 
rest  at  each  end  not  more  than  4"  on  any  wall,  and  shall  be  chamfered  at  each  end,  and 
-shall  have  a  double  rolock  arch  turned  over  said  timber  lintel;  arches  built  of  stone  or 


APPENDIX. 

brick  may  be  turned  over  openings  on  a  center,  which  may  be  struck  after  the  arch  is 
turned,  provided  the  arch  has  a  good  and  sufficient  rise,  and  that  the  piers  or  abutments 
are  of  sufficient  strength  to  bear  the  thrust  of  the  arch.  .  .  . 

§  17.  All  chimneys,  and  all  flues  in  stone  or  brick  walls,  in  any  building  hereafter 
erected,  altered,  or  repaired,  without  reference  to  the  purpose  for  which  they  may  be 
used,  shall  have  the  joints  struck  smooth  on  the  inside,  and  no  parging  mortar  shall  be 
used  on  the  inside ;  and  the  fire-backs  of  all  chimneys  hereafter  erected  shall  not  be  less- 
than  8"  in  thickness  ;  ...  no  wooden  furring  or  lath  shall  be  placed  against  any  flue,  metal 
pipe,  or  pipes  used  to  convey  heated  air  or  steam  in  any  building  ;  and  when  any  wall  shall 
hereafter  be  furred  or  lathed  with  wood,  the  space  between  the  lathing  and  wall  shall  be 
filled  with  plaster  between  the  top  and  underside  of  the  floor-beams  of  each  story,  so  as  to- 
prevent  fire  from  extending  from  one  floor  to  another.  And  no  air-flue  shall  be  used  at 
any  time  as  a  smoke-flue.  No  steam-pipe  shall  be  placed  within  2"  of  any  timber  or 
wood-work  as  aforesaid ;  when  the  said  space  of  2"  around  the  steam-pipe  is  objectionable, 
it  shall  be  protected  by  a  soap-stone  or  an  earthen  ring  or  tube.  No  base,  or  flooring,  or 
roofing,  or  any  other  wood- work  shall  be  placed  against  any  brick  or  other  flue  until  the 
same  shall  be  well  plastered  with  plaster-of-Paris  behind  such  wood- work.  .  .  . 

§  18.  No  smoke-pipe,  in  any  building  with  wooden  or  combustible  floors  and  ceilings,, 
shall  hereafter  enter  any  flue  unless  the  said  pipe  shall  be  at  least  18''  from  either  the 
floors  or  ceilings ;  and  in  all  cases  where  smoke-pipes  pass  through  stud  or  wooden 
partitions  of  any  kind,  whether  the  same  be  plastered  or  not,  they  shall  be  guarded  by 
either  a  double  collar  of  metal,  with  at  least  4"  air  space  and  holes  for  ventilation,  or  by  a. 
soap-stone  ring,  not  less  than  3"  in  thickness  and  extending  through  the  partition,  or  by  a 
solid  coating  of  plaster-of-Paris,  3"  thick,  or  by  an  earthenware  ring  3"  from  the  pipe.  .  .  . 

§  19.  In  no  building,  whether  the  same  be  a  frame  building  or  otherwise,  shall  any 
wooden  girders,  beams,  or  timbers  be  placed  within  12"  of  the  inside  of  any  flue,  whether 
the  same  be  a  smoke,  air,  or  any  other  flue.  All  wooden  beams  and  other  timbers  in  the 
party  wall  of  every  building  hereafter  to  be  erected  or  built,  of  stone,  brick,  or  iron,  shall 
be  separated  from  the  beam  or  timber  entering  in  the  opposite  side  of  the  wall  by  at  least 
8"  of  solid  mason- work.  No  floor-beam  shall  be  supported  wholly  upon  any  wood  par- 
tition, but  every  beam,  except  headers  and  tail-beams,  shall  rest,  at  one  end,  not  less  than 
4"  in  the  wall,  or  upon  a  girder,  as  authorized  by  this  act.  And  every  trimmer  or  header 
more  than  4'  long,  used  in  any  building  except  a  dwelling,  shall  be  hung  in  stirrup-irona, 
of  suitable  thickness  for  the  size  of  the  timbers.  .  .  . 

§  20.  In  all  buildings,  every  floor  shall  be  of  sufficient  strength  in  all  its  parts  to  bear 
safely  upon  every  superficial  foot  of  its  surface  75  Ibs. ;  and  if  used  as  a  place  of  public 
assembly,  120  Ibs. ;  and  if  used  as  a  store,  factory,  warehouse,  or  for  any  other  manufact- 
uring or  commercial  purposes,  from  150  to  500  Ibs.  and  upward  ;  and  every  floor  shall  be 
of  sufficient  strength  to  bear  safely  the  weights  aforesaid,  in  addition  to  the  weight  of  the 
materials  of  which  the  floor  is  composed  ;  and  every  column,  post,  or  other  vertical  sup- 
port shall  be  of  sufficient  strength  to  bear  safely  the  weight  of  the  portion  of  each  and 
every  floor  depending  upon  it  for  support,  in  addition  to  the  weight  required  as  above  to- 
be  supported  safely  upon  said  portions  of  said  floors.  In  all  calculations  fo^  the  strength 
of  materials  to  be  used  in  any  building,  the  proportion  between  the  safe  weight  and  the 
breaking  weight  shall  be  as  one  to  three  for  all  beams,' girders,  and  other  pieces  subjected 
to  a  cross-strain,  and  shall  be  as  one  to  six  for  all  posts,  columns,  and  other  vertical  sup- 
ports, and  for  all  tie-rods,  tie-beams,  and  other  pieces  subjected  to  a  tensile  strain.  And 
the  requisite  dimensions  of  each  piece  of  material  is  to  be  ascertained  by  computation  by 
the  rules  given  by  Tredgold,  Hodgkinson,  Barlow,  or  the  treatises  of  other  authors  now  or 
hereafter  used  at  the  United  States  Military  Academy  of  West  Point  on  the  strength  of 
materials,  using  for  constants  in  the  rules  only  such  numbers  as  have  been  deduced  from 
experiments  on  materials  of  like  kind  with  that  proposed  to  be  used.  ... 


670  APPENDIX. 

§  21.  In  all  fire-proof  buildings  hereafter  to  be  constructed,  where  brick  walls,  with 
wrought- iron  beams  or  cast  or  wrought  iron  columns  with  wrought-iron  beams,  are  used 
in  the  interior,  the  following  rules  must  be  observed : 

1.  All  metal  columns  shall  be  planed  true  and  smooth  at  both  ends,  and  shall  rest  on 
cast-iron  bed-plates,  and  have  cast-iron  caps,  which  shall  also  be  planed  true.     If  brick 
arches  are  used  between  the  beams,  the  arches  shall  have  a  rise  of  at  least  an  inch  and  a 
quarter  to  each  foot  of  space  between  the  beams. 

2.  Under  the  ends  of  all  the  iron  beams,  where  they  rest  on  the  walls,  a  stone  template 
must  be  built  into  the  walls  ;  said  templates  to  be  8''  wide  in  12"  walls,  and  in  all  walls  of 
greater  thickness  to  be  in  width  not  less  than  4"  less  than  the  width  of  said  walls,  and  not 
to  be,  in  any  case,  less  than  4"  in  thickness  and  18"  long.  .  .  . 

§  22.  All  exterior  cornices  and  gutters  of  all  buildings,  hereafter  to  be  erected  or  built, 
shall  be  of  some  fire-proof  material.  .  .  . 

§  23.  The  planking  and  sheathing  of  the  roof  of  every  building,  erected  or  built  as  afore- 
said, shall  in  no  case  be  extended  across  the  front,  rear,  side,  end,  or  party  wall  thereof, 
and  every  such  building,  and  the  tops  and  sides  of  every  dormer-window  thereon,  shall  be 
covered  and  roofed  with  slate,  tin,  zinc,  copper,  or  iron,  or  such  other  equally  fire- proof 
roofing.  .  .  . 


PATENT-OFFICE  DRAWINGS 

must  be  made  upon  pure  white  paper,  of  a  thickness  corresponding  to  three-sheet  Bristol 
board,  with  surface  calendered  and  smooth.  Indian  ink  alone  must  be  used. 

The  size  of  the  sheet  must  be  exactly  10  by  15  inches.  I"  from  its  edges  single  mar- 
ginal lines  are  to  be  drawn,  leaving  the  "  sight "  precisely  8"  by  13".  Within  this  margin 
all  work  must  be  included.  Measuring  downward  from  the  marginal  line  of  one  of  the 
shorter  sides,  a  space  of  not  less  than  1 J  inch  is  to  be  left  blank  for  the  heading  of  title, 
name,  number,  and  date. 

All  drawings  must  be  made  with  the  pen  only.  All  lines  and  letters  must  be  abso- 
lutely black,  clean,  sharp,  and  solid,  and  not  too  fine  or  crowded.  Surface  shading  should 
be  open,  and  used  only  on  convex  and  concave  surfaces  sparingly.  Sectional  shading 
should  be  made  by  oblique  parallel  lines,  which  may  be  about  ^"  apart. 

Drawings  should  be  made  with  the  fewest  lines  possible  con»istent  with  clearness. 
The  plane  upon  sectional  views  should  be  indicated  on  the  general  view  by  broken  or 
dotted  lines.  Heavy  lines  on  the  shade  sides  of  objects  should  be  used,  except  where  they 
tend  to  thicken  the  work  and  obscure  letters  of  reference;  light  to  come  from  the  upper 
left-hand  corner,  at  an  anp;le  of  45°. 

The  scale  of  the  drawing  to  be  large  enough  to  show  the  mechanism  without  crowd- 
ing ;  but  the  number  of  sheets  must  never  be  increased  unless  it  is  absolutely  necessary. 

Letters  and  figures  of  reference  must  be  carefully  formed,  and,  if  possible,  measure  at 
least  -J-"  in  height,  and  so  placed  as  not  to  interfere  with  a  thorough  comprehension  of  the 
drawing,  and  therefore  should  rarely  cross  the  lines.  Upon  shaded  surfaces  a  blank  space 
must  be  left  in  the  shading  for  the  letter.  The  same  part  of  an  invention  must  always 
"be  represented  by  the  same  character,  and  the  same  character  must  never  be  used  to 
designate  different  parts. 

The  signature  of  the  inventor,  by  himself  or  by  his  attorney,  is  to  be  placed  at  the 
lower  right-hand  corner  of  the  sheet,  and  the  signature  of  two  witnesses  at  the  lower  left- 
hand  corner,  all  within  the  marginal  line.  The  title  is  to  be  written  with  pencil  on  the 
back  of  the  sheet.  The  permanent  names  and  title  will  be  supplied  subsequently  by  the 
office  in  uniform  style. 

Drawings  should  be  rolled  for  transmission  to  the  office,  not  folded. 


APPENDi: 


671 


MENSURATION. 

Properties  of  Triangles. — It  has  been  already  shown  in  "  Geometrical  Problems  "  that 
to  construct  a  triangle  three  dimensions  must  be  known — the  three  sides,  or  two  sides  and 
the  included  angle,  or  one  side  and  the  two  adjacent  angles.  If  only  the  three  angles  are 
known,  triangles  of  varied  sizes  may 
be  constructed,  but  all  similar  to  each 
other.  To  determine  the  length  of  the 

side  of  a  right-angled  triangle  by  calcu-  A 

lation,  the  other  two  sides  being  known, 
use  these  formulae : 


CL 


A2  =  B2  +  C2,    or  A  =  VB2  +  C2 

B  =  4/Aa~-Tc27or  VA +~c~x~A"^~6 

C  =  VAa  —  B27  or    VA^B  x  A  — B.  FIG.  1. 

The  side  of  any  triangle  (Figs.  1,  2,  or  3)  can  be  found  by  the  following  formulae  : 

B  sin.  a 

— ,  and  consequently 


sin.  b 
B  sin.  a 


sm.  0  = 


A  sin.  5 

sm.a=  ___. 

The  area  of  a  triangle  is  equal  to  half  the  product  of  the 
base  by  the  height.  Taking  any  side  as  the  base,  say  B,  the 
height  is  readily  obtained  by  multiply- 
ing the  length  of  the  adjacent  side  A 
by  the  natural  sine  of  c.  All  figures 
bounded  by  straight  lines  can  be  divided 
into  triangles,  and  their  dimensions 
readily  calculated. 

Properties  of  circles. — The  circum- 
ference of  a  circle  is  equal  to  the  diam- 
eter multiplied  by  3-1416,  or  TT  (pi),  or 
approximately  3|. 

The  area  of  a  circle  is  equal  to  the  square  of  the  radius  multiplied  by  3'1416  (TT),  or  the 
square  of  the  diameter  multiplied  by  '7854. 

The  chord  A  0  (Fig.  4)  forms,  with  the  chords  of 
half  the  arc  and  the  three  radii,  right-angled  triangles 
whose  dimensions  may  be  calculated  as  given  above. 
But  the  solution  by  table  of  natural  sines  is  extremely 
simple ;  thus  the  chord  is  twice  the  sine  of  half  the 
angle  A  E  0  at  the  center  made  by  the  radii  to  the 
extremities  of  the  chord.  D  E  is  the  cosine  of  the 
angle  D  E  C  or  D  E  A,  and  the  versed  sine  B  D  is 
equal  to  radius  less  the  cosine. 

The  versed  sine  F  G  of  the  half  chord  is  equal  to 
about  one  quarter  of  the  versed  sine  D  B  of  the  whole 
chord. 

The  area  of  a  sector  A  B  0  E  is  to  that  of  the 
whole  circle  as  the  angle  at  the  center  A  E  0  is  to  360* 
the  length  of  the  arc  ABC,  will  give  the  area. 


FIG.  4. 
or  the  radius,  multiplied  bj  half 


672 


APPENDIX. 


The  length  of  an  arc  of  one  degree  =  radius  x  "017453. 
u         u       "    "     "     "     "    minute  =       u       x  '000291. 


second  = 


x  -000005. 


The  area  of  a  segment  A  B  0  D  is  equal  to  that  of  the  sector  less  the  area  of  the  tri- 
angle AEG  formed  by  the  chord  and  the  two  radii. 

To  find  the  circumference  of  an  ellipse,  divide  the  conjugate  or  short  diameter  by  the 
transverse  or  long  diameter,  and  find  the  quotient  in  the  first  column  in  the  accompanying 
table  ;  take  the  corresponding  number  from  the  table,  and  multiply  it  by  the  long  span. 

•2  =  2-10  -5  =  2-43  -8  =  2-84 

•3  =  2*20  -6  =  2  58  -9  =  2'99 

•4  =  2-30  -7  =  2-69  TO  =  3'14 

To  find  the  area  of  an  ellipse,  multiply  the  conjugate  by  the  transverse  diameter,  and 
the  result  by  '7854. 

The  area  of  a  parabola  is  the  product  of  the  base  by  two  thirds  the  height. 

Mensuration  of  Solids. — The  solidity  of  parallelopipeds,  cylinders,  and  prisms  is  found 
by  multiplying  the  base  by  the  altitude. 

The  solidity  of  cones  or  pyramids  is  found  by  multiplying  the  base  by  one  third  the 
vertical  height ;  of  frustums  of  pyramids,  the  sum  of  the  areas  of  the  two  ends  added  to 
the  square  root  of  their  product  multiplied  by  one  third  the  height. 

The  solidity  of  the  sphere  is  the  cube  of  the  diameter  multiplied  by  -5236. 

The  area  of  the  surface  is  the  square  of  the  diameter  multiplied  by  3*1416  (w),  or  four 
times  the  area  of  the  great  circle  passing  through  the  center. 

The  curved  surface  of  a  spherical  segment  is  the  product  of  the  diameter  of  the  sphere 
by  the  height  of  the  segment  by  3-1416. 

The  solidity  is  three  times  the  diameter  of  the  sphere,  less  twice  the  height  of  the  seg- 
ment, multiplied  by  the  square  of  the  height,  multiplied  by  -5236. 

The  solidity  of  the  wedge  is  the  length  of  the  edge  added  to  twice  the  length  of  the 
back,  multiplied  by  the  height  and  by  one  sixth  of  the  breadth  of  the  back. 


LINEAL  MEASUEE. 


Inches. 

Feet. 

Yards. 

Fath- 
oms. 

Links. 

Rods. 

Chains. 

Furlongs 

Statute 
miles. 

Nautical 
miles. 

Metres. 

1  — 

•08333 

•02778 

•0139 

•126 

•005 

•00126 

•000126 

•000016 

•0254 

12  = 

1 

•333 

•1667 

1-515 

•0606 

•0151 

•00151 

•00019 

.... 

0-3048 

36  = 

3 

1 

•5 

4-545 

•182 

•0454 

•00454 

•00057 

0-9144 

72  = 

6 

2 

1 

9-1 

•364 

•091 

•0091 

•00114 

.... 

1-8289 

7-92  = 

0-66 

•22 

•11 

1 

•04 

•01 

•001 

•000125 

.... 

•2012 

198  = 

16| 

5£ 

2f 

25 

1 

•25 

•025 

•003125 

.... 

5-0294 

792  = 

66 

22 

11 

100 

4 

1 

•10 

•0125 

.... 

20-118 

7920  = 

660 

220 

110 

1000 

40 

10 

1 

•125 

.... 

201-18 

63360  = 

5280 

1760 

880 

8000 

320 

80 

8 

1 

0-86755 

1609-41 

6086-07 

2028-69 

1-1527 

1 

1855-11 

39-3685  — 

3-2807 

1-0936 

•5468 

0-000621 

1 

Latin  prefixes,  as  milli-,  centi-,  deci-,  to  the  French  units  of  length  (metre),  surface  (are),  weight 
(gramme),  or  volume  (litre),  signify  YIOOO,  YIOO,  or  l/i0  of  the  unit;  as,  millimetre,  YIOOO  of  a  metre, 
decigramme,  l/i0  of  a  gramme.  Greek  prefixes,  as  kilo,  hekto,  deka,  multiples  of  the  unit  by  1,000. 
100,  or  10,  as  kilometre  =  1000  metres. 


APPENDIX. 


6Y3 


TABLE  OF  INCHES  AND  SIXTEENTHS  IN  DECIMALS  OF  A  FOOT. 


Inches. 

A 

A 

A 

A 

A 

A 

A 

A 

A 

H 

H 

w 

« 

« 

H 

0  

•ooo 

•005 

•010 

•016 

•021 

•026 

•031 

•036 

•042 

•047 

•052 

•057 

•062 

•068 

•073 

•078 

1  

•083 

•089 

•094 

•099 

•104 

•109 

•115 

•120 

•125 

•130 

•135 

•141 

•146 

•151 

•156 

•161 

2  

•167 

•172 

•177 

•182 

•187 

•193 

•198 

•203 

•208 

•214 

•219 

•224 

•229 

•234 

•240 

•245 

3  

•250 

•255 

•260 

•266 

•271 

•276 

•281 

•286 

•292 

•297 

•302 

•307 

•312 

•318 

•323 

•328 

4  

•333 

•339 

•344 

•349 

•354 

•359 

•365 

•370 

•375 

•380 

•385 

•391 

•396 

•401 

•406 

•411 

5  

•417 

•422 

•427 

•432 

•437 

•443 

•448 

•453 

•458 

•464 

•469 

•474 

•479 

•484 

•490 

•495 

6  

•500 

•505 

•510 

•516 

•521 

•526 

•531 

•536 

•542 

•547 

•552 

•557 

•562 

•568 

•573 

•578 

7.... 

•583 

•589 

•594 

•599 

•604 

•609 

•615 

•620 

•625 

•630 

•635 

•641 

•646 

•651 

•656 

•661 

8  

•667 

•672 

•677 

•682 

•687 

•693 

•698 

•703 

•708 

•714 

•719 

•724 

•729 

•734 

•740 

•745 

9.... 

•750 

•755 

•760 

•766 

•771 

•776 

•781 

•786 

•792 

•797 

•802 

•807 

•812 

•818 

•823 

•828 

10  

•833 

•839 

•844 

•849 

•854 

•859 

•865 

•870 

•875 

•880 

•885 

•891 

•896 

•901 

•906 

•911 

11.... 

•917 

•922 

•927 

•932 

•937 

•943 

•948 

•953 

•958 

•964 

•969 

•974 

•979 

•984 

•990 

•995 

MEASUKES  OF  SURFACE. 


Sq.  inches. 

Sq.  feet. 

Sq.  yards. 

Sq.  rods. 

Eoods. 

Acres. 

Sq.  miles. 

Sq.  metres. 

Ares. 

1  — 

'00694 

144  = 

1 

•Ill 

•0037 

.... 

.... 

.... 

•0929 

•0009 

1296  = 

9 

1 

•033 

.... 

.... 

.... 

•8361 

•0084 

.... 

272|r 

30J 

1 

•025 

•00625 

.... 

25-293 

0-253 

10890 

1210 

40 

1 

•25 

.... 

43560 

4840 

160 

4 

1 

•00156 

4046-86 

40-47 

.... 

27878400 

3097600 

.... 

.... 

640 

1 

.... 

25899 

1549-8  = 

10-763 

1-196 

•0395 

•0009 

•000247 

.... 

1 

•01 



1076-31 

119-60 





•02471 



100 

1 

MEASURES  OF  CAPACITY. 
LIQUID  MEASURE. 


Gills. 

Pints. 

Quarts. 

Gallons. 

Imp.  gallons. 

Litres. 

Cubic  feet. 

Cubic  in. 

Lbs.  water 
at  62°. 

1  = 

0-25 

0-125 

•03125 

•026 

•1183 

•0042 

7-219 

•26 

4  = 

1 

0-5 

0-125 

•1041 

•4731 

•01671 

28-875 

1-0412 

8  = 

2 

1 

0-25 

•2083 

0-9463 

•03342 

57-75 

2-0825 

32  = 

8 

4 

1 

0-8331 

3-7852 

0-1337 

231 

8-33 

38-4096  = 

9-6024 

4-8012 

1-2003 

1 

4-5435 

0-1605 

277-27 

10-00 

8-4534  = 

2-1133 

1-0567 

0-26417 

0-2201 

1 

0-0353 

61-0279 

2-2007 

239-36  = 

59-84 

29-92 

7'48 

6-232 

28-320 

1 

1728 

62-321 

'138528  = 

•034632 

•017316 

•004329 

•0036 

0-01639 

0-000579 

1 

•03606 

•01604 

27'727 

1 

674 


APPENDIX. 

DRY  MEASURE. 


Pints. 

Quarts. 

Gallons. 

Pecks. 

Bushels. 

1  = 

0-50 

0-125 

•0625 

0-01562 

2  = 

1 

0'25 

0-125 

0-0312 

8  = 

4 

1 

0-50 

0-125 

16  = 

8 

2 

1 

0-^5 

64  = 

32 

8 

4 

i 

The  standard  bushel  contains  2150-42  cubic  inches. 


WEIGHTS. 


APOTHECARIES'. 


TROY. 


Grains. 

Scruples. 

Drachms. 

Ounces. 

Pounds. 

1  = 

•05 

•0167 

•0021 

•00018 

20  = 

1 

•333 

•042 

•0035 

60  = 

3 

1 

•125 

•0104 

480  = 

24 

8 

1 

•083 

5760  = 

288 

96 

12 

1 

Grains. 

Pennyweights. 

Ounces. 

Pounds. 

1  = 

•042 

•0021 

•00018 

24  = 

1 

•05 

•0042 

480  = 

20 

1 

•083 

5760  = 

240 

12 

1 

AVOIRDUPOIS. 


Drachms. 

Ounces. 

Pounds. 

Hundred-weights  . 

Tons. 

French  grammes. 

1  = 

•0625 

•0039 

•000035 

•00000174 

1-771836 

16  = 

1 

•0625 

•000558 

•000028 

28-34938 

256  = 

16 

1 

•00893 

•000446 

453-59 

28672  = 

1792 

112 

1 

•05 

50802- 

673440  = 

35840 

2240 

20 

1 

1016041-6 

It  is  common  usage  here  to  omit  hundred-weights  (cwt.)  and  rate  tons  at  2,000  pounds  as  net,  and 
2240  Ibs.  as  gross. 


COMPAKISON  OF  WEIGHT. 


DYNAMIC  TABLE. 


Pounds 
apothecaries'. 

Pounds 
Troy. 

Pounds 
avoirdupois. 

Kilo- 
gramme. 

1  = 

1 

0-8229 

0-37324 

1  = 

1 

0-8229 

0-37324 

1-2153  = 

1-2153 

1 

0-4536 

2-6792  = 

2-6792 

2-2046 

1 

Pounds, 
fppt 

Kilogramme- 
metre. 

Horse- 
power. 

French 
horse-power. 

i  = 

0-13825 

•00003 

•000031 

7-2331  = 

1 

•000219 

•000222 

Per  min. 

33-000  = 

4562-3 

1 

1-01386 

32548-9  = 

4500 

0-98633 

1 

CUBIC  OR  SOLID  MEASURE. 


Cubic  inches. 

Cubic  feet. 

Cubic  yards. 

Cubic  metres. 

United  States  gallon. 

1  = 

•00058 

•000021 

•000016 

•004329 

1728  = 

1 

0-037 

0-0283 

7-48 

46656  = 

27 

1 

0-7646 

201-97 

61016  = 

35-31 

1-3078 

1 

264-141 

231  = 

0-1337 

•00495 

•00379 

1 

ss 


34  C 


53  «o  S 


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676  APPENDIX. 

WEIGHTS  OF  WROUGHT-IRON  AND  BRASS  PLATES  AND  WIRE,  SOFT  ROLLED-- 


BIRMINGHAM  GAUGE. 

No.  of 
gauge. 

AMERICAN   GAUGE. 

Plate  iron. 

Thickness  of 
each  number. 

Thickness 
of  each 
number. 

PLATKS   PER   SQUARE   FOOT. 

WIRE  PER  LINEAL  TOOT. 

Wrought 
iron. 

Brass. 

Wrought 
iron. 

Brass. 

Lbs. 

Inch. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

17-025 

•454 

0000 

•46 

17-25 

19-68 

•5607 

•6051 

15-9375 

•425 

000 

•4096 

15-361 

17-53 

•4447 

•4799 

14-25 

•38 

00 

•3648 

13-68 

15-61 

•3527 

•380& 

12-75 

•34 

0 

•3248 

12-182 

13-90 

•2797 

•8018 

11-25 

•3 

1 

•2893 

10-848 

12-38 

•2218 

•2393- 

10-65 

•284 

2 

•2576 

9-661 

11-02 

•1759 

•1898- 

9-7125 

•259 

3 

•2294 

8-603 

9-81 

•1395 

•1505 

8-925 

•238 

4 

•2043 

7-661 

8-74 

•1106 

•1193 

8-25 

•22 

5 

•1819 

6-822 

7-78 

•0877 

•0946 

7-6125 

•203 

6 

•1620 

6-075 

6-93 

•0695 

•0750 

6-75 

•18 

7 

•1442 

5-410 

6-17 

•0551 

•0595- 

6-1875 

•165 

8 

•1284 

4-818 

5-49 

•0437 

•0472. 

5-55 

•148 

9 

•1144 

4-291       . 

4-89 

•0347 

•0374 

6-025 

•134 

10 

•1018 

3-820 

4-36 

•0275 

•0296 

4-5 

•12 

11 

•0907 

3-402 

3-88 

•0218 

•0235 

4-0875 

•109 

12 

•0808 

3-030 

3-45 

•0173 

•0186- 

3-5625 

•095 

13 

•0719 

2-698 

3-07 

•0137 

•014& 

3-1125 

•083 

14 

•0640 

2-403 

2-74 

•0109 

•0117 

2-7 

•072 

15 

•0570 

2-140 

2-44 

•00863 

•00931 

2-4375 

•065 

16 

•0508 

1-905 

2-17 

•00684 

•00758 

2-175 

•058 

17 

•0452 

1-697 

1-93 

•00542 

•00585 

1-8375 

•049 

18 

•0403 

1-511 

1-72 

•00430 

•00464 

1-575 

•042 

19 

•0358 

1-345 

1-53 

•00341 

•00368 

1-3125 

•035 

20 

•(319 

1-198 

1-36 

•00271 

•00292. 

1-2 

•032 

21 

•0284 

1-067 

1-21 

•00215 

•00231 

1-05 

•028 

22 

•0253 

•9505 

1-08 

•00170 

•0018& 

•9375 

•025 

23 

•0225 

•8464 

•9660 

•00135 

•00145 

•825 

•022 

24 

•0201 

•7537 

•8602 

•00107 

•00115 

•75 

•02 

25 

•0179 

•6712 

•7661 

•00085 

•000916 

•675 

•018 

26 

•0159 

•5977 

•6822 

•000673 

•000726- 

•6 

•016 

27 

•0141 

•5323 

•6075 

•000534 

•000576 

•525 

•014 

28 

•0126 

•4740 

•5410 

•000423 

•0004  5  T 

•4875 

•013 

29 

•0112 

•4221 

•4818 

•000336 

•000362. 

•45 

•012 

30 

•0100 

•3759 

•4290 

•000266 

•000287" 

•375 

•01 

31 

•0089 

•3348 

•3821 

•000211 

•000228 

•3375 

•009 

32 

•0079 

•2981 

•3402 

•000167 

•000180- 

•3 

•008 

33 

•00708 

•2655 

•3030 

•000132 

•000143 

•2625 

•007 

34 

•00630 

•2364 

•2698 

•000105 

•000113 

•1875 

•005 

35 

•00561 

•2105 

•2402 

•0000836 

•00009015 

•15 

•004 

36 

•005 

•1875 

•214 

•0000662 

•0000715 

37 

•00445 

•1669 

•1905 

•0000525 

•00005671 

38 

•00396 

•1486 

•1697 

•0000416 

•00004  4  96- 

39 

•00353 

•1324 

•1511 

•0000330 

•0000356ft 

40 

•00314 

•1179 

•1345 

•0000262 

•00002827" 

Copper  is  about  5  per  cent  heavier  than  brass.     Lead  is  about  47  per  cent  heavier  than  wrought 
iron.     Zinc  is  about  7  per  cent  lighter  than  wrought  iron.     Sheet  copper  is  rated  by  weight  at  i 
many  ounces  per  square  foot,  and  sheet  lead  at  so  many  pounds  per  square  1 


APPENDIX. 


677 


TABLE  OF  DIMENSIONS  AND  WEIGHT  OF  WEOUGHT-1RON  WELDED  TUBES. 


Length  of  ;  Length  of 

VT—      —  * 

Nominal 
diameter. 

External 
diameter. 

Thick- 
ness. 

Internal 
diameter. 

Internal 
•  circum- 
ference. 

External 
circum- 
ference. 

pipe  per 
square 
loot  of 
internal 

pipe  per 
square 
foot  of 
external 

Internal 
area. 

Weight 
per  foot. 

no.  01 

threads 
per 
inch  of 

surface. 

surface. 

screw. 

Inches. 

Inches. 

Inches. 

Inches. 

Inches. 

Inches. 

Feet. 

Feet. 

Inches. 

Lbs. 

V. 

•40 

•068 

•27 

•85 

1-27 

14-15 

944 

•057 

•24 

27 

V* 

•54 

•088 

•36 

I'M 

1-7 

10-5 

7-075 

•104 

•42 

18 

3/8 

•67 

•091 

•49 

1-55 

2-12 

7-67 

5'657 

•192 

•56 

18 

v> 

•84 

.109 

•62 

1-96 

2'65 

6-13 

4-502 

•305 

•84 

14 

3A 

1-05 

.113 

•82 

2'59 

3'3 

4-64 

3-637 

•533 

1-13 

14 

i 

1-31 

•134 

1-05 

3-29 

4-13 

3-66 

2-903 

•863 

1-67 

iiVt 

1V4 

1-66 

•14 

1-38 

4-33 

5-21 

2-77 

2-301 

1-496 

2-26 

nV. 

W» 

1-9 

•145 

1-61 

5-06 

5-97 

2-37 

2-01 

2-038 

2-69 

ll1/*. 

2 

2-37 

•154 

2-07 

6-49 

7-46 

1-85 

1-611 

3-355 

3-67 

iiVt 

2/2 

2-87 

•204 

2-47 

7-75 

9-03 

1-55 

1-328 

4-783 

5-77 

8 

3 

3-5 

•217 

3-07 

9-64 

11- 

1-24 

1-091 

7-388 

7-55 

8 

»*/• 

4- 

•226 

3-55 

11-15 

12-57 

1-08 

0-955 

9-887 

9-05 

8 

4 

4-5 

•237 

4-07 

12-69 

14-14 

•95 

0-849 

12-73 

10-73 

8 

4V* 

5- 

•247 

4-51 

14-15 

15-71 

•85 

0-765 

15-939 

12-49 

8 

5 

5-56 

•259 

5-04 

15-85 

17-47 

•78 

0-629 

19-99 

14-56 

8 

6 

6-62 

•28 

6-06 

19-05 

20-81 

•63 

0-577 

28-889 

18-77 

8 

7 

7-62 

•301 

7-02 

22-06 

23-95 

•54 

0-505 

38-737 

23-41 

8 

8 

8-62 

•322 

7'98 

25-08 

27-1 

•48 

0-444 

50-039 

28-35 

8 

9 

9'69 

•344 

9- 

28-28 

30-43 

•42 

0-394 

63-633 

34-08 

8 

10 

10-75 

•366 

10-02 

31-47 

33-77 

•38 

0-355 

78-838 

40-64 

8 

Nominal 
diameter. 

Thickness, 
extra  strong. 

Thickness,  double 
extra  strong. 

Actual  inside  diameter. 
Extra  strong. 

Actual  inside  diameter. 
Double  extra  strong. 

Inches. 

Inches. 
O'lOO 

Inches. 

Inches. 
0-205 

Inches. 

l/. 

0-123 

0-294 

/  4 

0-127 



0-421 

v! 

0-149 

0-298 

0-542 

0-244 

3/4 

0-157 

0-314 

0-736 

0-422 

1 

0-182 

0-364 

0-951 

0-587 

I1/ 

0-194 

0-388 

1-272 

0-884 

I1/ 

0-203 

0406 

1-494 

1-088 

2 

0-221 

0442 

1-933 

1-491 

«V. 

0-280 

0-560 

2-315 

1-755 

3 

0-304 

0-608 

2-892 

2-284 

81/. 

0-321 

0-642 

3-358 

2-716 

4 

0-341 

0-682 

3-818 

3-136 

BOILER  TUBES. 


External 
diameter. 

Thickness, 
wire  gauge. 

Average 
weight. 

External 
diameter. 

Thickness, 
wire  gauge. 

Average 
Weight. 

Inches. 

No. 

Lbs.  per  foot. 

Inches. 

No. 

Lbs.  per  foot. 

W* 

16 

1- 

3 

11 

3-5 

i1/. 

15 

Me 

3J/4 

11 

4' 

!3/4 

14 

1-63 

4 

8 

6'4 

2 

13 

2- 

5 

7 

9-1 

2'/4 

12 

2-16 

6 

6 

12-3 

*v. 

12 

2-56 

7 

6 

15'2 

*"/I6 

11 

2-2 

8 

6 

16- 

678 


APPENDIX. 


HEAVY  PIPE  FOR  DRIVEN  WELLS. 

Tested  at  1200  pounds  hydraulic  pressure.     Furnished  in  five-foot  lengths. 


Size  (inches)  

H 

H 

2 

'    2J 

O 

3* 

4 

Weight  per  foot,  Ibs.. 

3-62 

2-75 

3-75 

6-00 

7-75 

9-25 

11-00: 

HEAVY  WROUGHT  GALVANIZED  IRON  SPIRAL  RIVETED  PIPES, 

WITH  FLANGED  CONNECTIONS. 
Tested  at  150  pounds  hydraulic  pressure.     Regalvanized  after  riveting. 


Inside  diameter  (inches)   . 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

Wire  gauge,  Nos  

20 

20 

20 

18 

18 

18 

18 

16 

16 

16 

Nominal  weight  per  foot,  Ibs.  .  . 

2i 

4 

5 

6 

7 

8 

9 

12 

13 

14 

Manufactured  lengths,  20  feet  or  less.     Elbows  and  other  fittings,  cast  iron. 
LIGHT  PIPE,  SUITABLE  FOB  HOUSE  LEADERS,  VENTILATING,  AIR,  AND  BLOWER  PIPES,  ETC. 


Inside  dia,meter  (inches)       .    ... 

2 

2A 

3 

3i 

4 

4-i- 

5 

5i 

6 

Nominal  wei'rht  per  foot  Ibs  . 

§ 

f 

1 

1 

14- 

If 

H 

14 

1£ 

TABLE  OF  COPPER  AND  BRASS  RODS  ONE  FOOT  IN  LENGTH. 

To  find  the  weight  of  copper  or  brass  pipe,  take  the  weight  of  the  exterior  diameter  from  the 
table,  and  subtract  from  it  the  weight  of  a  rod  equal  to  that  of  the  interior  diameter,  or  bore. 


Diamet'r 
in 
inches. 

Copper. 

Brass. 

Diamefr 
in 
inches. 

Copper. 

Brass. 

DiametT 
in 
inches. 

Copper. 

Brass. 

'/• 

•047 

•045 

W* 

7-993 

7-593 

4'A 

55-62 

52-27 

8A« 

•106 

•101 

1"/16 

8-630 

8-198 

43/8 

58-94 

5539 

'A 

•189 

,     -179 

I'A 

9-270 

8-806 

4V* 

62-36 

58-60 

6A« 

•296 

•281 

!13Ae 

9-950 

9-452 

45/e 

65-87 

61-90 

% 

•426 

•405 

iVs 

10-642 

10-110 

43/4 

69-48 

6 

V» 

•579 

•550 

I"/,. 

11-370 

10-801 

47/8 

73-19 

68-77 

V. 

•757 

•719 

2 

12-108 

11-503 

5 

77-43 

72-76 

9A« 

•958 

•910 

*'/• 

13-668 

12-985 

5'/8 

80-89 

76-00' 

5/8 

1-182 

1-123 

2'A 

15-325 

14-559 

5V4 

84-88 

79-76 

"Ae 

1-431 

1-360 

23/s 

17-075 

16-221 

53/8 

88-97 

83-60 

8A 

1-703 

1-618 

2'A 

18-916 

17-970 

5'/i 

93-15 

87-63 

13A« 

1-998 

1-898 

25/e 

20-856 

19-808 

55/s 

97-44 

91-56 

V- 

2-318 

2-202 

23/4 

22-891 

21-746 

53/4 

101-81 

95-68 

15Ae 

2-660 

2-527 

r/. 

25-019 

23-768 

5Y« 

106-29 

99-88 

1 

3-027 

2-876 

3 

27-243 

25-881 

6 

110-85 

104-15 

i'A. 

3-417 

3-246 

3'/s 

29-559 

28-081 

6!A 

12030 

113-04 

W« 

3-831 

3-639 

ffif* 

31-972 

30-373 

6V, 

130-10 

122-26 

i3A« 

4-269 

4-056 

33/e 

34-481 

32-757 

63/4 

140-32 

131-85 

i'A 

4-723 

4-487 

31/, 

37-081 

35-227 

7 

150-86 

141-76 

l'/i. 

5-214 

4-953 

35/e 

39-777 

37-788 

v!A 

161-87 

152-10 

13A 

5-723 

5-437 

*/« 

42-568 

40-440 

»«/• 

173-22 

162-77 

!'/«. 

6-255 

5-943 

r/, 

45-455 

43-182 

73/4 

184-97 

173-81 

1% 

6-811 

6-470 

4 

48-433 

46-000 

8 

197-03 

185-14 

!9Ae 

7-390 

7-020 

4V« 

52-40 

49-24 

APPENDIX. 


6?$ 


NUMBER  OF  BURDEN'S  RIVETS  IN  ONE  HUNDRED  POUNDS. 


Lengths. 

DIAMETER. 

Lengths. 

•JS.  B. 

i 

1 

H 

} 

1 

1,092 

665 

.... 

.... 

5 

90 

| 

1,027 

597 

.... 

.... 

6| 

85 

1 

940 

538 

450 

.... 

6 

80 

H 

840 

512 

415 

.... 

6| 

75 

U 

797 

487 

389 

356 

7 

70 

If 

760 

460 

370 

329 

n 

67 

H 

730 

440 

357 

280 

8 

65 

if 

711 

420 

340 

271 

H 

61 

if 

693 

390 

325 

262 

9 

57 

H 

648 

375 

312 

257 

9i 

54 

2 

608 

360 

297 

243 

10 

51 

H 

573 

354 





10! 

47 

2i 

555 

347 

280 

232 

H 

525 

335 

260 

220 

2f 

500 

312 

242 

208 

3 

460 

290 

224 

197 

H 

433 

267 

212 

180 

3| 

413 

248 

201 

169 

H 

395 

241 

192 

160 

4 

.... 

230 

184 

158 

4* 

.... 

220 

177 

150 

4I 

.... 

210 

171 

146 

4£ 

.... 

200 

166 

138 

5 

.... 

190 

161 

135 

BJ 

.... 

180 

156 

130 

B* 

.... 

172 

151 

124 

5f 

.... 

164 

145 

120 

6 

.... 

157 

140 

115 

6i 

.... 

150 

138 

111 

6£ 

.... 

146 

134 

107 

6f 

.... 

143 

129 

104 

7 



140 

125 

100 

WROUGHT  SPIKES  — NUMBER  TO  A  KEG  OF  ONE  HUNDRED  AND  FIFTY  POUNDS. 


LENGTH. 

i" 

A" 

1" 

A" 

i" 

Inches, 
3   

2,250 

3i!r 

,890 

1,208 

4 

,650 

1,135 

4-i  . 

,464 

1,064 

5  

,380 

930 

742 

6  

,292 

868 

570 

7  

,161 

662 

482 

445 

306 

8  

635 

455 

384 

256 

9    

573 

424 

300 

240 

10  ... 

391 

270 

222 

11  

249 

203 

12.. 

236 

180 

680  APPENDIX. 

LENGTHS  OF  CUT  NAILS  AND  SPIKES,  AND  NUMBER   IN  A  POUND. 


Size. 
3d. 

Length. 

No. 

Size. 

Length. 

No. 

Size. 

Length. 

No. 

Inches. 
1± 

420 

Sd. 

Inches. 
2* 

100 

30d 

Inches. 
4 

24 

4 

1* 

270 

10 

3 

65 

40 

4± 

20 

5 

If 

220 

12 

8* 

52 

60 

6 

2 

175 

20 

3| 

28 

WEIGHTS  OF  LEAD   PIPE  PEE  FOOT  IN  LENGTH. 


Caliber. 

MASK. 

AAA 

AA 

A 

B 

C 

D 

E 

Lbs.  oz. 
0     2 

0  10 
0     9$ 

1     8 

i 
i 

I 

A 

i 

f 
t 

i 

li 
1* 

if 

2 

H 

3 

8* 

4 

4* 
5 
6 

Lbs.    oz. 

Lbs.    oz. 

Lbs.    oz. 

Lbs     oz. 

Lbs.  oz. 

Lbs.  oz. 

Lbs.  oz. 

0     2 

1     12 

1       5 

1        2 

1       0 

0  14 

0    7 

3     .. 
2       8 
3       8 
4     14 
6     .. 
6     12 
8       0 

2       0 

1      10 

1       3 

1     0 

0  10 
0  12 
1     4 
1     3 
2     4 
2     8 
3     8 
3     0 
3  10 
4     0 
3     0 

0  12 
1     0 
2     0 
2     0 
2     0 

2     12 
3       3 
4       8 
5     12 

7      0 

2       8 
3       0 
4       0 
4     11 
6       4 

2       0 
2       3 
3       4 
3     11 
5       0 

1     7 
1  12 
2     8 
3     0 
4     4 

10     11 

8       8 
8     14 

6       7 
7      0 

5       0 
6       0 

4     0 
5     0 

THICKNESS. 

WASTE. 

1 

A 

i 

4 

A 

16     11 
19       9 
22       8 
25       6 

*31       3 

13     10 
16       0 
18       7 
20     14 

10     10 
12       9 
14       8 

16       7 
18       6 
20       5 

7       3 
9       4 
10     12 
12       2 
13       9 
15       0 

6     0 
5     0 

4     0 

3     8 

8     0 

6     0 

10     8 
10    8 
12     0 

7     6 

TABLE  OF  THE  WEIGHT  OF  A  CUBIC  FOOT  OF  WATER  AT  DIFFERENT  TEM- 
PERATURES. 


Fahren- 
heit. 

Centi- 
grade. 

Weight  in 
pounds. 

Fahren- 
heit. 

Centi- 
grade. 

Weight  in 
pounds. 

Fahren- 
heit. 

Centi- 
grade. 

Weight  in 
pounds. 

Degrees. 
32 

Degrees. 
0 

62-42 

Degrees. 
95 

Degrees. 
35 

62-06 

Degrees, 
167 

Degrees. 

75 

60-87 

39 

4 

62-42 

104 

40 

61-95 

176 

80 

60-68 

41 

5 

62-42 

113 

45 

61-83 

185 

85 

60-48 

50 

10 

62-41 

122 

50 

61-69 

194 

90 

60-27 

59 

15 

62-37 

131 

55 

61-55 

203 

95 

60-04 

68 

20 

62-32 

140 

60 

61-39 

212 

100 

59-83 

77 

25 

62-25 

149 

65 

61-23 

86 

30 

62-16 

158 

70 

61-06 

APPENDIX. 


631 


PROPERTIES  OF  SATURATED  STEAM, 
FROM  "RICHABDS'S  STEAM-ENGINE  INDICATOR,"  BY  CHAS.  T.  PORTER. 


ELASTIC 

HEAT,    IN   DEGREES 

$i 

1 

ELASTIC 

HEAT,   IN   DEGREES 

I-        1 

FORCE. 

FAHRENHEIT. 

«  I 

«    ^ 

FORCE. 

FAHRENHEIT. 

•-  I 

*S 

.—  ~ 

V     * 

£  ? 

'"=>  * 

«    B 

sr 

1     I   - 

1* 

§  M 

£•*      * 

k   • 

si 

£? 

c  8 

% 

»! 

l\ 

c   S 

•s  3 

"o   5 

!? 

*\ 

a  *" 

!) 

II 

i 

Jj 

I1 

—     V 

fi 

\\ 

"c  ~ 

J  5 
If 

s 

V 

1* 

I3 

r 

i 

2-04 

102- 

1043-0 

1145-0 

•0029 

•037 

64 

130-40 

296-9 

907-6 

1204-5 

•1416  j  1-754 

2 

4-08 

126-3 

1026-1 

1152-4 

•0057 

•071      65 

132-44 

298-0 

906-8 

1204-8 

•1436      -779 

3 

6-11 

141-6 

1015-4 

1157-1 

•0084 

•104      66 

134-48 

299-0 

906-1 

1205-1 

•1456    1-804 

4 

815 

153-1 

1007-5 

1160-6 

•0110 

136 

67 

136-51 

300-0 

905-4 

1205-4 

•14V6    1-829 

5 

10-19 

162-3 

1001-0 

1163-4 

•0135 

•167 

68 

138-55 

300-9 

904-8 

1205-7 

•1496 

•854 

6 

12-22 

1701 

995-6 

1165-8 

•0160 

198 

69 

140-59 

301-9 

904-1 

1206-0 

•1516 

•879 

7 

14-26 

176-9 

990-9 

1167-9 

•0185 

•228   !  70 

142-63 

302-9 

903-4 

1206-3 

•1536       -904 

8 

16-30 

182-9 

.  986-7 

1169-7 

•0209 

•258      71 

144-66     303-9 

902-7 

1206-6 

•1556     1-929 

9 

18-34 

188-3 

983-0 

1171-3 

•0233 

•238      72 

146-70     304  8 

902-1 

1206-9 

•1576    1-954 

10 

20-38 

193-2 

979-6 

1172-8 

•0257 

•318  ,    73 

148-74 

305-7 

901-5 

1207-2' 

1596    1-979 

11 

22-41 

197-8 

976-4 

1174-2 

•0281 

•348      74  ! 

150-78 

306-6 

900-9 

120Y-5 

•1616 

2-004 

12 

24-45 

201-0 

973-5 

1175-5 

•0304 

•377      75 

152-81    307-5 

900-3 

1207-8 

•1636 

2-029 

13 

26-48 

205-9 

970-8 

1176-7 

•0327 

•406 

76 

154-85     308-4 

899-6 

1208-0 

•1656 

2-054 

14 

28-53     209-6 

968-2 

1177-8 

•0350 

•435 

77 

156-89     309-3 

899-0 

1208-3 

1676 

2-079 

14-7 

atmos. 

78 

158-93 

310-2 

898-4 

1208-6 

1696 

2-103 

15 

30-56 

213-0 

965-8 

1178-9 

•0373 

•463 

79 

160-96 

311-1 

897-8 

1208-9 

•1716    2-127 

16 

32-60 

216-3 

963-6 

1179-9 

•0396 

•492 

80 

163-00 

312-0 

897-1 

1209-1 

•1736 

2-151 

17 

34-64 

219-4 

961-5 

1180-9 

•0419 

•520      SI 

165-04 

312-8 

896-6 

1209-4 

•1756 

2-175 

18 

36-68 

222-4 

959-4 

1181-8 

•0442 

•548 

J  82 

167-08 

313-6 

896-1 

1209-7 

•1776 

2-199 

19 

38-71 

225-2 

957-5 

1182-7 

•0465 

•576 

i  83   169-11 

314-5 

895-4 

1209-9 

1795 

2-223 

20 

40-75 

228-0 

955-5 

1183-5 

•0487 

•604 

84 

171-15 

315-3 

894-8 

1210-1 

1814 

2-247 

21 

42-79 

230-6 

953-7 

1184-3 

•0510 

•632 

85   173-19 

316-1 

894-3 

1210-4 

•1833 

2-271 

22 

44-83 

233-1 

951-9 

1185-0 

•0532 

•660 

86   175-23 

316-9 

893-8 

1210-7 

1852 

2-295 

23 

46-86 

235-5 

950-2 

1185-7 

•0554 

•688 

87  177-26 

317-8 

893-1 

1210-9 

•1871 

2-319 

24 

43-90 

237-9 

948-6 

1186-5 

•0576 

•715      88  179-30 

318-6 

892-5 

1211-1 

•1891 

2-343 

25 

50-94 

240-2 

947-0 

1187-2 

•0598 

•742   1  89  1181-34 

319-4 

892-0 

1211-4 

•1910 

2-367 

26 

52-98 

242-3 

945-6 

1187-9 

•0620 

•769  !    90  183-38 

320-2 

891-4 

1211-6 

•1930 

2-391 

27 

55-01 

244-4 

944-1 

1188-5 

•0642 

•796      91   185-41 

321-0 

890-8 

1211-8 

•1950 

2-415 

28 

57-05 

246-4 

942-7 

1189-1 

•0664 

•823   1   92   187-45 

321-7 

890-3 

1212-0 

•1970 

2-439 

29 

•59-09 

248-4 

941-3 

1189-7 

•0686 

•850      93   189-49 

322-5 

889-8 

1212-3 

•1990 

2-463 

30 

61-13 

250-4 

939-9 

1190-3 

•0707 

•877 

94 

191-53 

323-3 

889-2 

1212-5 

•2010 

2-487 

31 

63-16 

252-3 

938-5 

1190-8 

•0729 

•904 

1   95  ! 

193-56 

324-1 

888-7 

1212-8 

•2030 

2-511 

32 

65-20 

254-1 

937-3 

1191-4 

•0751 

•931  |    96  i 

195-60 

324-8 

888-2 

1213-0 

•2050 

2-535 

33 

67-24 

255-9 

936-1 

1192-0 

•0772 

•958      97 

197-64 

325-6 

887-7 

1213-3 

•2070 

2-559 

34 

69-28 

257-6 

934-9 

1192-5 

•0794 

•985 

1  98  J199-68 

326-3 

887-2 

1213-5 

•2089 

2-583 

35 

71-31 

259-3 

933-7 

1193-0 

•0815 

1-012 

99   201-71 

327-1 

886-6 

1213-7 

•2108 

2-607 

36 

73-35 

260-9 

932-6 

1193-5 

•0837 

1-033    100  203-75 

327-8 

886-1 

1213-9 

•2127 

2-631 

37 

75-39 

262-6 

931-4 

1194-0 

•0853 

1-064    101  1205-79 

328-5 

885-7 

1214-2 

•2147 

2-655 

38 

77-43 

264-2 

930-3 

1194-5 

•0879 

1-090    102  1207-88 

329-2 

885-2 

1214-4 

•2167 

2-679 

39 

79-46 

265-8 

929-2 

1195-0 

•0900 

1-116     103 

209-86 

329-9 

884-7 

1214-6 

•2186 

2-703 

40 

81-50 

267-3 

923-1 

1195-4 

•0921 

1-142  1  104 

211-90 

330-6 

884-2 

1214-8 

•2205 

2-727 

41 

83-54 

268-7 

927-2 

1195-9 

•0942 

1-168    105  1213-94 

331-3 

883-7 

1215-0 

•2224 

2-751 

42 

85-58 

270-2 

926-1 

1196-3 

•0963 

1-194    106 

215-98 

331-9 

883-3 

1215-2 

•2243 

2-775 

43 

87-61 

271-6 

925-2 

1196-8 

•0983 

1-220  i  107 

218-01 

332-6 

882-8 

1215-4 

•2262 

2-799 

44 

89-65 

273-0 

924-2 

1197-2 

•1004 

1-246    108 

220-05 

333-3 

882-3 

1215-6 

•2281 

2-823 

45 

91-69 

274-4 

923-2 

1197-6 

•1025 

•272    109 

222-09 

334-0 

881-8 

1215-8 

•2300 

2-847 

46 

93-73 

275-8 

922-2 

1198-0 

1046 

•298     110 

224-13 

334-6 

881-4 

1216-0 

•2319 

2-871 

47 

95-76 

277-1 

921-3 

1198-4 

•1067  !     -324    111 

226-16 

335-3 

880-9 

1216-2 

•2337 

2-895 

48 

97-80 

278-4 

920-4 

1198-8 

•1087       '350     112 

228-20 

336-0 

880-4 

1216-4 

•2355 

2-919 

49 

99-84 

279-7 

919-5 

1199-2 

•1108 

•376    113 

230-24 

336-7 

879-9 

1216-6 

•2374 

2-943 

50 

101-88    281-0 

918-6 

1199-6 

•1129  |     -402    114 

232-28 

337-4 

879-4 

1216-8 

•2392 

2-967 

51 

103-91 

282-3 

917-7 

1200-0 

•1150 

1-428    115  i 

234-31  . 

338-0 

879-0 

1217-0 

•2410 

2-990 

52 

105-95 

283-5 

916-9 

1200-4 

1171 

1-454    116   236-35 

338-6 

878-6 

1217-2 

•2428 

3-013 

53 

107-99    284-7 

916-1 

1200-8 

1192 

1-479    117  238-39 

339-3 

878-1 

1217-4 

•2446 

3-036 

54 

110-03    285-9 

915-2 

1201-1 

•1212 

1-504    118  ,240-43 

339-9 

877-7 

1217-6 

•2465 

3-059 

55 

112-06    287-1 

914-4 

1201-5 

•1232 

1-529    119  !-242-46 

340-5 

877-3 

1217-8 

•2484 

3-082 

56 

114-10    288-2 

913-6 

1201-8 

•1252     1-554    120   244-50 

341-1 

876-9 

1218-0 

•2503 

3-105 

57 

116-14    289-3 

912-9 

1202-2 

1272  i  1-579    121 

246-54 

341-8 

876-4 

1218-2 

•2522 

3-130 

58 

113-18    290-4 

9121 

12025 

•1293 

1-604    122  248-58 

342-4 

876-0 

1218-4 

•2541 

3-155 

59 

120-21     291-6 

911-3 

1202-9 

1314 

1-629    123 

250-61 

343-0 

875-6 

1218-6 

•2560 

3179 

60 

122-25    292-7 

910-5 

1203-2 

•1335 

1-654    124 

252-65 

343-6 

875-1 

1218-7 

•2579 

3-203 

61 

124-29    293-8 

909-8 

1203-6 

•1356 

1-679     125 

254-69 

344-2 

874-7 

1218-9 

•2598 

3-227 

62  126-33     294-8 

909-1       1203-9 

•1376 

1-704    126 

256-73     344-8 

874-3 

12191 

•2617 

3-251 

63  1128-36    295-9 

908-3       1204-2 

1396 

1-729     127 

258-76    345-4 

873-9 

1219-3 

•2636 

3-275 

682 


APPENDIX. 


PEOPEETIES   OF  SATUEATED   STEAM— (Continued.) 


ELASTIC 

HEAT,    IN   DEGREES 

l| 

| 

ELASTIC 

HEAT,    IN   DEGREES 

J 

1 

FORCE. 

FAHRENHEIT. 

§    rS 

<U      Q 

FORCE. 

FAHRENHEIT. 

|l 

IJ 

S" 
Jo 

o  e 

i. 

£  o. 

D     O 

§-3 

°  "S 

ft 

Id 

e  a 

k 

C     * 

l| 

•si 

.1 

ft 

*  2 

li 

P 

II 

1 

I1 

I1 

II 

ft 

CO 

jl 

|  J 

"*     3 

jl 

I1 

-  « 

0   J 

I- 

I1 

128   260-80 

346-0 

873-4 

1219-4 

•2655 

3-299 

140 

285-25 

352-9 

868-6 

1221-5 

•2883 

3-582 

129   262-84 

346-6 

873-0 

1219-6 

•2674 

3-323 

141 

287-29 

353-4 

868-3 

1221-7 

•2902 

3-605 

130   264-88 

347-2 

872-6 

1219-8 

•2693 

3-347 

142 

289-33 

354-0 

867-9 

1221-9 

•2921 

3-628 

131    266-91 

347-8 

872-2 

1220-0 

•2712 

3-371 

143 

291-36 

354-5 

867-5 

1222-0 

•2940 

3-651 

132 

268-95 

348-3 

871-9 

1220-2 

•2731 

3-395 

144 

293-40 

355-0 

867-2 

1222*2 

•2959 

3-674 

133 

270-99 

348-9 

871-5 

1220-4 

•2750 

3-419 

145 

295-44 

355-6 

866-8 

1222-4 

•2978 

3-697 

134   273-03 

349-5 

871-1 

1220-6 

•2769 

3-443 

146 

297-48 

356-1 

866-4 

1222-5 

•2997 

3-720 

135   275-06 

350-0 

870-7 

1220-7 

•2788 

3-467 

147 

299-51 

356-7 

866-0 

1222-7 

•3016 

3-74S 

136    277-10 

350-6 

870-3 

1220-9 

•2807 

3-490 

148 

301  -55 

357-2 

865-7 

1222-9 

•3035 

3-765 

137    279-14 

351-2 

869-8 

1221-0 

•2826 

3-513 

149 

303-59 

357-8 

865-2 

1223-0 

•3054 

3-787 

138    281-18 

351-8 

869-4 

1221-2 

•2845 

3-536 

150 

305-63 

358-3 

864-9 

1223-2 

•3073 

3-809 

139 

283-21 

352-3 

869-1 

1221-4 

•2864 

3-559 

1 

TABLE  OF  MEAN  PEESSUEES  IN  STEAM  CYLINDEES   AT  DIFFEEENT  BATES  OF 

EXPANSION. 


Portion  of 
stroke  during 
which  steam 

Mean  press- 
ure during 
whole  of 

Portion  of 
stroke  during 
which  steam 

Mean  press- 
ure during 
whole  of 

Portion  of 
stroke  during 
which  steam 

Mean  press- 
ure during 
whole  of 

Portion  of 

stroke  during 
which  steam 

Mean  press- 
ure during 
whole  of 

is  admitted. 

stroke. 

is  admitted. 

stroke. 

is  admitted. 

stroke. 

is  admitted. 

stroke. 

•80 

•98 

•56 

•88 

•40 

•77 

•24 

•58. 

•77 

•97 

•54 

•87 

•38 

•75 

•22 

•55- 

•74 

•96 

•52 

•86 

•36 

•73 

•20 

•52 

•70 

•95 

•50 

•85 

•34 

•71 

•18 

•49- 

•68 

•94 

•48 

•83 

•32 

•68 

•16 

•45 

•66 

•93 

•46 

•82 

•30 

•66 

•14 

•42 

•62 

•92 

•44 

•80 

•28 

•64 

•12 

•37 

•60 

•90 

•42 

•78 

•26 

•61 

•10 

•33 

•58 

•89 

Examples  of  Application  of  above  Table. — To  find  the  mean  pressure  in  a  condensing 
engine  with  an  initial  pressure,  as  shown  by  the  gauge,  of  75  pounds,  and  a  cut-off  at  -20, 
or  £  stroke. 

The  actual  initial  pressure  above  0  is  75  +  15,  or  90  pounds.  Mean  pressure  at  '20  cut- 
off in  table  '52  for  each  pound  of  initial  pressure,  90  x  '52  =  46-8  mean  pressure  above  0 
in  cylinder  ;  but  as  the  vacuum  in  the  cylinder  can  never  be  perfect,  an  allowance  of  two 
to  three  pounds  is  to  be  made;  46'8  —  2'8  =  44,  which  may  be  taken  as  the  probable 
actual  mean  pressure  to  be  used  in  estimating  the  H.  P.  or  Ibs.  ft.  of  work  of  the  engine — 
made  up  thus :  Mean  pressure  x  area  of  steam  piston  in  square  inches,  less  \  that  of  the 
piston-rod  x  length  of  stroke  in  feet  x  number  of  strokes  per  minute  =  Ibs.  ft.  of  work 
per  minute,  and  divided  by  33,000  =  II.  P. 

If  the  engine  is  non-condensing,  then  the  deduction  from  the  mean  pressure  would  be 
the  whole  atmospheric  pressure,  14/7,  and  probably  about  1*3  back  pressure,  or  say,  16 
pounds,  and  the  mean  effective  pressure  in  the  cylinder  would  be  for  the  cut-off  and  initial 
power  as  above,  46*8  —  16,  or  30'8  pounds. 

In  estimating  for  the  per  cent  of  cut-off  or  steam  follow,  the  clearances  are  to  be  esti- 
mated with  the  stroke  and  cut-off. 

It  may  often  be  convenient  to  estimate  the  amount  of  water  and  coal  necessary  for  an 
engine,  which  can  be  done  approximately  by  taking  the  tension  or  pressure  of  the  steam 
at  any  part  of  the  stroke  after  the  cut-off,  finding  in  table  the  weight  of  one  cubic  foot 


APPENDIX.  68a 

of  steam  corresponding  to  this  pressure,  and  multiplying  it  by  the  number  of  cubic  feet 
in  the  cylinder  at  the  point  taken,  which  will  be  the  weight  of  steam  used  per  stroke. 
Multiplying  this  product  by  the  number  of  strokes  per  working  day,  will  give  the  total 
weight  of  water  used  as  steam  ;  and  if  8  pounds  of  steam  be  allowed  for  each  pound  of  coal, 
it  will  give  a  fair  average  of  tho  coal  consumption  during  working  hours.  There  will  be 
additional  coal  used  for  getting  up  steam  or  for  banking,  and  more  water  will  be  used  than 
shown  by  the  steam  in  the  cylinder,  as  there  will  be  water  entrained  with  the  steam,  and 
condensed  in  passages  and  cylinder,  equal  to  25  per  cent  more,  say,  in  total,  10  pounds  of 
water  for  each  pound  of  coal  fed  on  the  grates. 


THE  FLOW   OF   WATER. 

The  velocity  of  water  in  a  stream  or  channel  is  often  taken  approximately  by  floats 
along  different  threads  of  the  current.  If  the  channel  be  an  artificial  one  of  rectangular 
section,  the  average  velocity  may  be  determined  very  nearly  by  a  number  of  such  experi- 
ments, with  a  tube  float,  extending  nearly  to  the  bottom  of  the  channel ;  but  in  the  rivers 
and  streams,  if  surface  floats  be  used,  allowance  is  to  be  made  for  the  friction  of  water  on 
the  bed  of  the  stream,  and  want  of  uniformity  in  the  flow.  There  are  a  variety  of  tachom- 
eters to  determine  the  velocities  beneath  the  surface,  and  to  afford  data  for  averages. 

In  the  flow  of  water  through  apertures  the  theoretic  velocity  in  feet  per  second  is 
8'04  ^ h,  h  being  the  head  or  height  of  surface  of  water  in  feet  above  the  center  of  the 
aperture.  But  in  all  apertures  the  discharge  is  less  than  the  product  of  the  area  of  their 
section  by  the  theoretic  velocity.  There  are  contractions  which  reduce  the  effective  sec- 
tion. If  the  discharge  be  through  a  thin  plate  into  air,  in  which  the  contractions  are 
around  the  entire  periphery,  the  discharge  is  T67  of  that  due  to  the  section  and  theoretic 
velocity.  If  the  edges  are  rounded,  or  the  discharge  be  through  a  short  pipe  or  ajutage, 
or  beneath  the  surface  of  the  water,  the  loss  is  less,  and  by  suitable  ajutages  it  may  be 
almost  entirely  eliminated. 

For  the  common  purpose  of  gauging  or  determining  the  discharge  of  large  pumps  or 
small  streams,  the  most  accurate  measure  is  by  weirs,  on  which  many  experiments  have 
been  made,  but  those  of  Mr.  James  B.  Francis,  G.  E.,  which  are  embodied  in  "  Lowell 
Hydraulic  Experiments,"  embrace  a  more  practical  range  than  any  other,  and  are  consid- 
ered standard. 

His  general  formula,  on  which  the  following  table  is  calculated,  is  Q  =  3'33  (I  —  -2A)  Af, 
in  which  Q  is  the  discharge  in  cubic  feet  per  second,  I  the  length  of  the  weir,  and  h  the 
height  of  water  above  the  crest  of  the  weir,  both  in  feet;  h  is  taken  either  at  the  side  of 
the  weir  or  a  slight  distance  up  stream ;  usually,  a  pipe  with  small  perforations  is  laid 
parallel  with  the  weir,  on  the  bottom,  and  connected  with  a  tight  vertical  box,  in  which 
the  oscillations  of  the  water  surface  are  reduced  to  a  mean. 

In  the  table,  the  discharge  is  given  for  one  foot  in  length ;  but  as  in  weirs  there  are 
usually  two  end  contractions,  virtually  reducing  the  length,  and  met  in  the  formula  above 
by  —  -2&,  a  column  of  correction  has  been  added,  which  is  to  be  subtracted  from  the 
product  of  discharge,  as  given  in  the  other  columns  of  the  table,  by  the  length  in  feet. 

Example. — Let  the  weir,  with  end  contractions,  be  5-3  feet  long,  and  depth  of  waterv 
or  h  =  0-612. 

By  table  the  discharge  for  one  foot  in  length  is 1  -594 

5-3 


8-4482 
Correction -196 


Discharge  in  cubic  feet  per  second 8-252, 


684 


APPENDIX. 


DISCHARGE,  IN  CUBIC  FEET  PER    SECOND,  OF  A  WETR  ONE  FOOT  LONG,  WITH- 
OUT CONTRACTION  AT  THE  ENDS;    FOR  DEPTHS  FROM  0-500  TO  0-999  FEET. 


Correction 
for  con- 
tractions. 

Depth. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

•012 

0-20 

0-298 

0-300 

0-302 

0-305 

0-307 

0-309 

0-311 

0-314 

0-316 

0-318 

•013 

•21 

0-320 

0-323 

0-325 

0-327 

0-330 

0-332 

0-334 

0-337 

0-339 

0-341 

•015 

•22 

0-344 

0-346 

0-348 

0-351 

0-353 

0-355 

0-358 

0-360 

0-362 

0-365 

•017 

•23 

0-367 

0-370 

0-372 

0-374 

0-377 

0-379 

0-382 

0-384 

0-387 

0-389 

•019 

•24 

0-391 

0-394 

0-396 

0-399 

0-401 

0-404 

0-406 

0-409 

0-411 

0-414 

•021 

•25 

0-416 

0-419 

0-421 

0-424 

0-426 

0-429 

0-431 

0-434 

0-436 

0-439 

•023 

•26 

0-441 

0-444 

0-447 

0-449 

0-452 

0-454 

0-457 

0-459 

0-462 

0-465 

•025 

•27 

0-467 

0-470 

0-472 

0-475 

0-478 

0-480 

0-483 

0-485 

0-488 

0-491 

'028 

•28 

0-493 

0-496 

0-499 

0-501 

0-504 

0-507 

0-509 

0-512 

0-515 

0-517 

•030 

•29 

0-520 

0-523 

0-525 

0-528 

0-531 

0-534 

0-536 

0-539 

0-542 

0-544 

•033 

0-30 

0-547 

0-550 

0-553 

0-555 

0-558 

0-561 

0-564 

0-566 

0-569 

0-572 

•036 

•31 

0-575 

0-577 

0-580 

0-583 

0-586 

0-589 

0-591 

0-594 

0-597 

0-600 

•039 

•32 

0-603 

0-606 

0-608 

0-611 

0-614 

0-617 

0-620 

0-623 

0-625 

0-628 

•042 

•33 

0-631 

0-634 

0-637 

0-640 

0-643 

0-646 

0-649 

0-651 

0-654 

0-657 

•045 

•34 

0-660 

0-663 

0-666 

0-669 

0-672 

0-675 

0-678 

0-681 

0-684 

0-687 

-048 

•35 

0-689 

0-692 

0-695 

0-698 

0-701 

0-704 

0-707 

0-710 

0-713 

0-716 

-052 

•36 

0-719 

0-722 

0-725 

0-728 

0-731 

0-734 

0-737 

0-740 

0-743 

0-746 

-056 

•37 

0-749 

0-752 

0-755 

0-759 

0-762 

0-765 

0-768 

0-771 

0-774 

0-777 

-059 

•38 

0-780 

0-783 

0-786 

0-789 

0-792 

0-795 

0-799 

0-802 

0-805 

0-808 

•063 

•39 

0-811 

0-814 

0-817 

0-820 

0-823 

0-827 

0-830 

0-833 

0-836 

0-839 

-067 

0-40 

0-842 

0-846 

0-849 

0-852 

0-855 

0-858 

0-861 

0-865 

0-868 

0-871 

•072 

•41 

0-874 

0-877 

0-881 

0-884 

0-887 

0-890 

0-893 

0-897 

0-900 

0-903 

•076 

•42 

0-906 

0-910 

0-913 

0-916 

0-919 

0-923 

0-926 

0-929 

0-932 

0-936 

•081 

•43 

0-939 

0-942 

0-945 

0-949 

0-952 

0-955 

0-959 

0-962 

0-965 

0-969 

•085 

•44 

0-972 

0-975 

0-978 

0-982 

0-985 

0-988 

0-992 

0-995 

0-998 

1-002 

•090 

•45 

1-005 

1-009 

1-012 

1-015 

1-019 

1-022 

1-025 

1-029 

1-032 

1-035 

-095 

•46 

1-039 

1-042 

1-046 

1-049 

1-052 

1-056 

1-059 

1-063 

1-066 

1-070 

•100 

•47 

1-073 

1-076 

1-080 

1-083 

1-087 

1-090 

1-094 

1-097 

1-100 

1-104 

-106 

•48 

1-107 

1-111 

1-114 

1-118 

1-121 

1-125 

1-128 

1-132 

1-135 

1-139 

-111 

•49 

1-142 

1-146 

1-149 

1-153 

1-156 

1-160 

1-163 

1-167 

1-170 

1-174 

-118 

0-50 

1-177 

1-181 

1-184 

1-188 

1-191 

1-195 

1-199 

1-202 

1-206 

1-209 

•124 

•51 

1-213 

1-216 

1-220 

1-223 

1-227 

1-231 

1-234 

1-238 

1-241 

1-245 

-130 

•52 

1-249 

1-252 

1-256 

1-259 

1-263 

1-267 

1-270 

1-274 

1-278 

1-281 

•136 

•53 

1-285 

1-288 

1-292 

1-296 

1-299 

1-303 

1-307 

1-310 

1-314 

1-318 

•143 

•54 

1-321 

1-325 

1-329 

•332 

1-336 

1-340 

1-343 

1-347 

1-351 

1-355 

-150 

•55 

1-358 

1-362 

1-366 

•369 

1-373 

1-377 

1-381 

1-384 

1-388 

1-392 

•157 

•56 

1-395 

1-399 

1-403 

•407 

1-410 

1-414 

1-418 

1-422 

1-425 

1-429 

•164 

•57 

1-433 

1-437 

1-441 

•444 

1-448 

•452 

1-456 

1-459 

1-463 

1-467 

•171 

•58 

1-471 

1-475 

1-478 

•482 

1-486 

•490 

1-494 

1-498 

1-501 

1-505 

-178 

•59 

1-509 

1-513 

1-517 

•521 

1-524 

•528 

1-532 

1-536 

1-540 

1-544 

-186 

0-60 

1-548 

1-551 

1-555 

•559 

1-563 

•567 

1-571 

1-575 

1-579 

1-583 

•194 

•61 

1-586 

1-590 

1-594 

•598 

1-602 

•606 

1-610 

1-614 

1-618 

1-622 

•202 

•62 

1-626 

1-630 

1-633 

•637 

1-641 

•645 

1-649 

1-653 

1-657 

1-661 

•210 

•63 

1-665 

1-669 

1-673 

•677 

1-681 

•685 

1-689 

1-693 

1-697 

1-701 

•218 

•64 

1-705 

1-709 

1-713 

•717 

1-721 

•725 

1-729 

1-733 

•737 

1-741 

-227 

•65 

1-745 

1-749 

1-753 

•757 

1-761 

•765 

1-769 

1-773 

•777 

1-781 

'236 

•66 

1-785 

1-790 

1-794 

•798 

1-802 

•806 

1-810 

1-814 

•818 

1-822 

•245 

•67 

1-826 

1-830 

1-834 

•838 

1-843 

•847 

1-851 

1-855 

•859 

1-863 

•254 

•68 

1-867 

1-871 

1-875 

1-880 

1-884 

•888 

1-892 

1-896 

•900 

1-904 

•263 

•69 

1-909 

1-913 

1-917 

1-921 

1-925 

1-929 

1-934 

1-938 

•942 

1-946 

"273 

0-70 

1-950 

1-954 

1-959 

1-963 

1-967 

1-971 

1-975 

1-980 

1-984 

1-988 

•283 

•71 

1-992 

1-996 

2-001 

2-005 

2-009 

2-013 

2-017 

2-022 

2-026 

2-030 

-293 

•72 

2-034 

2-039 

2-043 

2-047 

2-051 

2-056 

2-060 

2-064 

2-068 

2-073 

-303 

•73 

2-077 

2-081 

2-085 

2-090 

2-094 

2-098 

2-103 

2-107 

2-111 

2-115 

-314 

•74 

2-120 

2-124 

2-128 

2-133 

2-137 

2-141 

2-146 

2-150 

2-154 

2-159 

'324 

•75 

2-163 

2-167 

2-172 

2-176 

2-180 

2-185 

2-189 

2-193 

2-198 

2-202 

APPENDIX. 


685- 


DISCHARGE,   IN   CUBIC  FEET  PER  SECOND,   OF  A   WEIR  ONE   FOOT  LONG,   WITH- 
OUT CONTRACTION  AT   THE  ENDS;  FOR   DEPTHS  FROM  0-500  TO  0-999   FEET. 

(  Continued. ) 


Correction 
for  con- 
tractions. 

Depth. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

•335 

•76 

2-206 

2-211 

2-215 

2-219 

2-224 

2-228 

2-232 

2-237 

2-241 

2-246 

•346 

•77 

2-250 

2-254 

2-259 

2-263 

3-267 

2-272 

2-276 

2-281 

2-285 

2-290 

•358 

•78 

2-294 

2-298 

2-303 

2-307 

2-312 

2-316 

2-320 

2-325 

2-329 

2-334 

•369 

•79 

2-238 

2-343 

2-347 

2-351 

2-356 

2-360 

2-365 

2-369 

2-374 

2-378 

•381 

0-80 

2-383 

2-387 

2-392 

2-396 

2-401 

2-405 

2-410 

2-414 

2-419 

2-423 

•393 

•81 

2-428 

2-432 

2-437 

2-441 

2-446 

2-450 

2-455 

2-459 

2-464 

2-468 

•406 

•82 

2-473 

2-477 

2-482 

2-486 

2-491 

2-495 

2-500 

2-504 

2-509 

2-513 

•413 

•83 

2-518 

2-523 

4-527 

2-532 

2-536 

2-541 

2-545 

2-550 

2-554 

2-559 

•431 

•84 

2-564 

2-568 

2-573 

2-577 

2-582 

2-587 

2-591 

2-596 

2-600 

2-605 

•444 

•85 

2-610 

2-614 

2-619 

2-623 

2-628 

2-633 

2-637 

2-642 

2-646 

2-651 

•457 

•86 

2-656 

2-660 

2-665 

2-670 

2-674 

2-679 

2-684 

2-688 

2-693 

2-698 

•470 

•87 

7-702 

2-707 

2-712 

2-716 

2-721 

2-726 

2-730 

2-735 

2-740 

2-744 

•484 

•88 

2-749 

2-754 

2-758 

2-763 

2-768 

2-772 

2-777 

2-782 

2-786 

2-791 

•498 

•89 

2-796 

2-801 

2-805 

2-810 

2-815 

2-819 

2-824 

2-829 

2-834 

2-838 

•512 

0-90 

2-843 

2-848 

2-853 

2-857 

2-862 

2-867 

2-872 

2-876 

2-881 

2-886 

•526 

•91 

2-891 

2-895 

2-900 

2-905 

2-910 

2-915 

2-919 

2-924 

2-929 

2-934 

•541 

•92 

2-938 

2-943 

2-948 

2-953 

2-958 

2-963 

2-967 

2-972 

2-977 

2-982 

•555 

•93 

2-986 

2-991 

2-996 

3-001 

3-006 

3-011 

3-015 

3-020 

3-025 

3-030 

•570 

•94 

3-035 

3-040 

3-044 

3-049 

3-054 

3-059 

3-064 

3-069 

3-074 

3-078 

•586 

•95 

3-083 

3-088 

3-093 

3-098 

3-103 

3-108 

3-113 

3-117 

3-122 

3-127 

•601 

•96 

3-132 

3-137 

3-142 

3-147 

3-152 

3-157 

3-162 

3-166 

3-171 

3-176 

•677 

•97 

3-181 

3-186 

3-191 

3-196 

3-201 

3-206 

3-211 

3-216 

3-221 

3-226 

•632 

•98 

3-231 

3-235 

3-240 

3-245 

3-250 

3-255 

3-260 

3-265 

3-270 

3-275 

•648 

•99 

3-280 

3-285 

3-290 

3-295 

3-300 

3-305 

3-310 

3-315 

3-320 

3-325 

Flow  of  Water  through  Pipes. — Figs.  5  and  6  are  diagrams  showing,  by  inspection, 
the  million  gallons  delivered  in  24  hours  under  varying  resistance-heads  or  sines  of 

slopes  ( —  j  of  c.ean  cast-iron  pipes  of  diameters  from  6"  to  36".     They  are  calculated 

from  the  table  of  velocities  in  J.  F.  Fanning's  "  Practical  Treatise  on  Hydraulics  and 
Water-Supply  Engineering.'11 

Illustration  of  the  Application  of  Diagram. — To  determine  the  million  gallons  dis- 
charged per  24  hours  through  a  12"  pipe  with  '02  sine  of  slope.  The  intersection  of  the 
horizontal  of  -02  by  the  curve  of  12"  is  on  the  ordinate  4  millions,  which  will  be  dis- 
charge to  be  determined. 

Again,  to  determine  the  loss  of  head  per  foot  in  length  of  a  30"  pipe  in  delivering  25 
million  gallons  per  24  hours.  The  intersection  of  the  ordinate  of  25  millions  with  the  30" 
curve  is  in  the  horizontal  '0067,  the  loss  of  head  to  be  determined. 

It  will  be  seen  that  a  36"  pipe  would  deliver  the  same  quantity  with  a  loss  of  but  '0025 
feet  per  foot  in  length. 

These  diagrams  are  applicable  to  long  mains  with  a  uniform  current. 

Flow  through  Sewers. — Fig.  7  is  a  diagram  similar  to  the  preceding,  by  which  may 
be  readily  determined  the  cubic  feet  per  second  that  would  flow  through  circular  sewers 
from  12"  to  72"  diameter,  with  various  falls  of  from  TT5*Jff  to  y^  of  a  foot  per  foot  of 
length. 

It  is  calculated  by  the  formula  given  by  A.  Fteley,  M.  A.  S.  0.  E.,  in  the  description  of 
the  "  Additional  Supply  from  Sudbnry  River,"  for  the  Boston  Water-Works,  and  deduced 
from  experiments  made  by  him  on  those  works. 


686 


APPENDIX. 


2QE 


i 

9) 

i 


FIG.  5. 


APPENDIX. 


1 


GALLONS  //v  24- novas 

FIG.  6. 


688 


APPENDIX. 


^^£1 


-s-- 


± 


\| 


Fio.  7. 


APPENDIX. 


689 


The  formula  is — 


in  which  V  =  velocity  in  feet  per  second, 

C  =  coefficient  varying  with  R,  as  given  in  the  following  table, 

area 

R  =  hydraulic  mean  radius  = : , 

wetted  perimeter 

which  in  circular  sewers  is  =  J  of  the  diameter. 

total  fall 

I  =  sine  of  inclination  =  —          — -  . 
total  length 


R 

C 

R 

C 

R 

C 

o-i 

96-3 

0-6 

119-4 

1-1 

128-5 

0-2 

104-7 

0-7 

121-7 

1-2 

129-8 

0-3 

109-9 

0-8 

123-6 

1-3 

131-1 

0-4 

113-8 

0-9 

125-4 

1-4 

132-2 

0-6 

116-9 

1-0 

127-0 

1-5 

133-3 

Example. — To  determine  the  cubic  feet  per  second  that  would  be  discharged  by  a 
sewer  4'  or  48"  diameter  with  a  fall  per  foot  of  -006. 

The  intersection  of  the  horizontal  '006  with  the  48  in.  curve  is  on  the  ordinate  124,  which 
is  the  quantity  per  second  which  would  be  discharged  under  the  conditions  of  the  example. 

On  the  other  hand,  to  determine  the  fall  per  foot  necessary  to  give  a  60"  sewer  to  dis- 
charge 200  cubic  feet  per  second.  Following  up  the  ordinate  200  to  its  intersection  of 
the  60"  curve,  its  intersection  will  be  found  on  the  -0049  horizontal,  which  will  be  the  fall 
required. 

For  the  same  cubic  feet  of  discharge  per  second,  it  will  be  seen  by  the  diagram  that  a 
72"  sewer  would  require  but  -0018  fall  per  foot,  and  a  54"  sewer,  for  the  same  discharge, 
a  fall  of  -0086  feet. 

Flow  of  Gas  through  Cast-iron  Mains. — The  usual  formula  found  in  hand-books  is 


Q  =  1350 


HD 
GL* 


in  which  Q  =  cubic  feet  per  hour,  D  diameter,  and  H  head  of  water-pressure,  both  in 
inches,  L  length  of  pipe  in  yards,  and  Gr  specific  gravity  of  the  gas ;  if  the  last  be  taken  at 
•42,  L  at  1  mile  or  1,760  yards,  and  H  one  inch,  then  Q  =  1200,  D  *  and 


D  =  Vl?440,000  Q2  =  17-25  Qf . 

It  will  be  observed  that,  in  the  flow  through  the  pipes,  equivalent  sections  do  not  imply 
equal  discharges ;  that,  by  the  formula  above,  the  flow  through  4  pipes  under  the  same 
head  is  not  equal  to  that  of  one  pipe  of  double  the  diameter,  but  that  the  flow  is  as  the 
square  root  of  the  5th  power  of  the  diameter  (D*). 

Flow  of  Air  through  Pipes. — B.  F.  Sturtevant  &  Co.,  in  the  appplication  of  their 
fans  and  connections,  found  it  very  convenient  to  have  tables  of  the  value  of  pipes  of  dif- 
ferent diameters  in  conveying  air  under  different  pressures,  and  the  practical  economy  in 
this  application  in  the  matter  of  power  for  the  transmission  of  air.  On  the  following 
page  are  the  tables  published  by  them  of  the  results  of  their  calculations. 


44 


690 


APPENDIX. 


IN 

TABLE  FOR  EQUALIZING   THE   DIAMETER   OF  PIPES. 

1 

i 

P 

arties  putting  up  blast  pipes  are 
four  6-inch  pipes  is  the  same  as 

very  liable  to  think,  because  the  combined  area  of 
one  12-inch  pipe,  that  the  four  pipes  will  convey  the 

2 

5.7|     2 

3 

16 

2.7 

3 

4 

32 

5.7 

2. 

* 

8 

ame  quantit 
as  it  acti 
have 

rn 

y  of  air  with  the  same  ease  and  freedom  that  the  12-inch  will,  where- 
lally  does  take  5-7—  almost  six  6-inch  pipes.    Again,  16  3-inch  pipes 
the  combined  area  of  one  12-inch  pipe,  but  in  actual  practice  it  takes 
ust  32  3-inch  pipes  to  do  the  work  of  one  12-inch. 

5 

56 

9.8|     3.6 

-8|    5 

6 

88 

1C 

5.7 

.8|  1.6 

6 

7 

129 

23  J     8.3|      .1|  2.3 

l.S 

8 

180 

32 

12  J      .7    3.2 

2.1 

|1.4 

IM 

8^             This  is  d 
•s|  »1             the  s 

ae 
m 
Ph 

tot 

all  i 
e  la 
an 

1 

het 

ipes 

•ge 
lete 
Tl 

xce 
ovc 
Eigu 
rsir 

16  fl 

111 

ss  of  friction  fo 
r  that  in  the  lar 
-es  at  the  top  ol 
inches  of  the  b 
gures  at  the  int 
ic  with  the  ve 
pipes,  of  the 
of  the  co 

r  every  cubic  foot  of  air  in 
ge- 
each  column  give  the  di- 
ranch  pipes, 
ersection  of  the  horizontal 
rtical  give  the  number  of 
diameter  given  at  the  top 
umn,  that  will  be  equal  in 
city  for  conveying  air  to 
>ne  given  opposite  in  the 
first  column. 

9 

244 

42 

16 

.6|  4.3)  2.8 

10 

317 

56 

20  |      .9|  5.7 

3.6!  2.4| 

.-7|  1.3 

lo]  r 

11 

402 

71 

26 

2 

7.0|  4.5J  3.1| 

.2|  1.7 

12 

501 

88  |     32 

1C 

9.0[  B.Tj  3.8J 

.8    2.0J   l.C|   1.2|  12] 

13 

613 

107 

39   |   19 

11 

6.9 

4.7 

.4    2.5|   1.9f  1.5|   l.?|  13 

14 

737 

129 

47 

23 

13 

8.3|5.7| 

.1    3.0|  2.3|   1.8|   1.5| 

.5 

15 

876 

152 

56 

27 

16 

9.9|   6.7| 

.8|  3.6|  2.8|  2.2|   1.8| 

.4|   1.2 

15 

16 

1026 

180  |     65 

32 

18  |   11 

7.9| 

.7    4.2 

3.2J   2'Gi   M 

.' 

i    1"« 

1.2 

16 

17 

1197 

208 

76 

37 

21 

13 

9.2 

.6    4.9|  3.8|   2.9(  2.4| 

|    1.6 

1.4 

1.2 

17  1            capa 

18 

1375 

239  |     88 

43 

24 

1C 

10  | 

.7    5.7 

4.8|  3.4[  2.8| 

.3|   1.9|  l.G|   1.3 

|  l.2|  18]               < 

19 

1580 

275 

.100 

49 

28  |   18 

12 

6.5|  5.   |  3.9|   3.2| 

.G 

|2.2|   1.8)1.5 

1.3|  1.2I19) 

20 

1797 

313 

114 

56 

32 

20  i  14 

.9    7.4 

5.7|  4.5;.3.6| 

.9|2.5|2.1|1.7 

1.5|   1.3[  1.1|20 

22 

2284 

398 

145 

71 

41 

26  |  18  |  13      9.3 

7.2J   5.7J  4.5| 

.7|  3.l|   2.G|  2.2|   P.9|   1.7|  1.4|  1.3 

22  | 

•  2|24| 

24 

2834 

493  |   180 

88 

(0 

32  |   22  |   16.      12 

8.9|   7.6|   5.7| 

.6|  3.8|3.2 

26 

3474 

605 

219  |108 

G2 

39 

27  |  19  |  14 

11   |  8.6[  6/9| 

.7 

|4.7 

4.0)  3.4 

2.9|   2.5|  2.2)   1.9 

•  5|   1.2|26| 

28 

4165 

725 

2G5 

129 

74 

48 

32      23  |   17  |   13  |   10  |  8.3| 

:8|  5.7 

4.8    4.1 

3.5|  3.0|  2.C|  2.3 

.S|  i.s|  i.2|28| 

30 

4963 

864 

315  |154 

88 

50 

38      28      20  |  16  |   12  |  9.3| 

,OJ  6.7 

5.7 

4.7 

4.1|  3.G|  3.0J  2.6 

.2|    1.7|   1.4|1.2|3O| 

36 

7818 

361 

497 

243 

139 

83 

60      43      32 

25  |   19  |  16  | 

3 

11 

8.9 

7.6i  C.5|  5.7J  5.0|  4.3 

.4|  2.7|  2.2|1.!>|1.G|36| 

42    J11488  J2000 

730  [358  |205 

129 

8*      63      47 

36   |  29  |  23  | 

9 

16 

13 

11   |  9.G|  8.5|   7.3|  6.4 

.0]   4.1|  3.3|2.8;2.3|l.5|42] 

48    J15989  |2792  J1081  J492 

232 

180 

123      88      66 

50  |  39   |  32  |  26  |  22      18 

16  |   13  1   12  |   10  |  8.9 

.0|  5.7,'  4.7;3.8,3.2|2.l|l.4|48| 

64    J21560  |3753  |l3C8 

671   |384  |244 

166    119      88 

68  |   53  |  43  |  35  |  29 

24 

21 

18  |   16  |   15  |  12 

.4|   7.6|   G.2j5.  2[4.  3'2.8|l.9|l.  3)54 

6O    |27913  |4879  |l781 

872  (499 

314 

215   J154     115 

88  |  69  |  56   |  46 

38 

Si 

27 

23   |  20  |   18  |   16 

12   |   9.9|   8.li6.7;5.7;3.8(2.4|l.8|l.S 

DIAMETER  OF  PIPES  IN  INCHES. 

LOSSES  OF  PRESSURE  PER  100  FEET  MUST  BE  PROVIDED  FOR  BY  EXTRA  SPEED  AND  POWER  ON  THIS 

BLOWER. 


Hi 

?j& 

LOSS  OF  PRESSURE  IN  OUNCES  PER  SQUARE  INCH. 

1  inch. 

2  inch. 

3  inch. 

4  inch. 

6  inch. 

8  inch. 

10  inch. 

12  inch. 

14  inch 

16  inch 

18  inch 

20  inch 

22  inch 

100 

•on 

•006 

•004 

•003 

•002 

•001 

•001 

•001 

•001 

•001 

•001 

.-001 

•001 

200 

•044 

•022 

•015 

•on 

•007 

•006 

•004 

•004 

•003 

•003 

•002 

•002 

•002 

400 

•178 

•088 

•059 

•044 

•030 

•022 

•018 

•015 

•013 

•on 

•010 

•009 

•008 

600 

•400 

•200 

•133 

•100 

•067 

•050 

•040 

•033 

•029 

•025 

•022 

•020 

•018 

800 

•711 

•356 

•237 

•178 

•119 

•089 

•071 

•059 

•051 

•044 

•040 

•036 

•032 

1000 

1-111 

•556 

•370 

•278 

•185 

•139 

•111 

•092 

•079 

•069 

•062 

•056 

•051 

1200 

1-600 

•800 

•533 

•400 

•267 

•200 

•160 

•133 

•114 

•100 

•089 

•080 

•073 

1400 

2-178 

1-089 

•726 

•544 

•363 

•282 

•218 

•181 

•156 

•136 

•121 

•109 

•099 

1600 

2-844 

1-422 

•948 

•711 

•474 

•356 

•284 

•237 

•203 

•178 

•158 

•142 

•129 

1800 

3-600 

1-800 

1-200 

•900 

•600  |     '450 

•360 

•300 

•257 

•225 

•200 

•180 

•164 

2000 

4-444 

2-222 

1-481 

1-111       -741 

•556 

•444 

•370 

•317 

•278 

•247 

•222 

•202 

2200 

5-378 

2-689  i   1-793 

1-344 

•896 

•672 

•538 

•448 

•384 

•336 

•299 

•269 

•244 

2400 

6-400 

3-200     2-133 

1-600 

1-067 

•800 

•640 

•533 

•457 

•400 

•356 

•320 

•291 

2600 

7-511 

3-756 

2-504     1-877 

1-252 

•939 

•751 

•626 

•537 

•468 

•417 

•376 

•341 

2800 

8-711 

4-356 

2-904     2-178 

1-452 

1-089 

•871 

•726 

•622 

•544 

•484 

•436 

•396 

3000 

10-000 

5-000 

3-333     2-500 

1-667 

1-250 

1-000 

•833 

•714 

•625 

•556 

•500 

•455 

3200 

11-378 

5-689 

3-792     2-844 

1-896 

1-422 

•138 

•948 

•813 

•711 

•632 

•569 

•517 

3400 

12-844 

6-422 

4-281     3-211 

2-141 

1-606 

•284 

1-070 

•917 

•827 

•714 

•642 

•584 

3600    14-400 

7-200 

4-800     3-600 

2-400 

1-800 

•440 

1-200 

1-029 

•900 

•800 

•720 

•655 

3800  !  16-044 

8-022 

5-349 

4-011 

2-674 

2-006 

•604 

1-337 

1-146 

1-003  |   -891 

•802 

•729 

4000 

17-778 

8-889 

5-926 

4-444 

2-963 

2'222 

•778 

1-481 

1-270 

1-111 

•988 

•889 

•808 

4400 

10-705 

7-175 

5-353     3-569 

2-676 

2-141 

1-784 

1-537 

1-344 

1-189  1-071 

•973 

4800 

12-800 

8-533 

6-400     4-267 

3-200 

2-560 

2-133 

1-829 

1-600  1-422 

1-280 

1-164 

5200 

15-022 

10-015 

7-511      5-007 

3-756 

3-004 

2-504 

2-146 

1-871    1-670 

1-502 

1-366 

5600 

17-422 

11-615 

8-711     5-807 

4-356 

3-484  i   2-904 

2-489 

2-178 

1-936 

1-742 

1-584 

6000 

20-000 

13-333 

10-000     6-667 

5-000 

4-000  i   3-333 

2-857 

2-500 

2-222  2-000 

1-818 

APPENDIX. 


691 


TABLES  OF  THE  CIRCUMFERENCES  OF  CIRCLES  TO  THE  NEAREST  FRACTION  OF 
PRACTICAL  MEASUREMENT  ;  ALSO,  THE  AREAS  OF  CIRCLES,  IN  INCHES  AN1> 
DECIMAL  PARTS,  LIKEWISE  OF  FEET  AND  DECIMAL  PARTS. 


Circumfer- 

Diameter 

Area 

Area 

Circumfer- 

Diameter 

Area 

Area 

ence  in  feet 

in 

in  square 

in  square 

ence  in  feet 

in 

in  square 

in  square 

and  inches. 

inches. 

inches. 

feet. 

and  inches. 

inches. 

inches. 

feet. 

1     61- 

6 

28-27 

•196 

•20 

iV 

•003 

i    H 

6* 

29-46 

•204 

•39 

1 

•012 

1    71 

6* 

30-68 

•212 

•59 

"1% 

•028 

1     8 

6f 

31-92 

•220 

•78 
•98 

i 

•049 
•077 

1     8f 

6* 
6| 

33-18 
34-47 

•228 
•237 

1-18 

f 

•110 

1     9i 

6| 

35-78 

•246 

1-37 

T2? 

•150 

1     9| 

6^ 

37-12 

•256 

1-57 

* 

•196 

1  10 

7 

38-48 

•267 

1-77 

A 

•248 

1  10| 

7-| 

39-87 

•277 

1-96 

f 

•307 

1  10| 

*7* 

41-28 

•287 

2-16 

•371 

1   11* 

71 

42-72 

•297 

2-36 

f 

•442 

1   11* 

7* 

44-18 

•307 

2-55 

•518 

1  11* 

7f 

45-66 

•318 

2-75 

f 

•601 

2     Of 

71 

47-17 

•328 

2-94 

•690 

2     0| 

7* 

48-71 

•338 

•3* 

•785 

•0054 

2     1* 

8 

50-26 

•349 

3* 

i. 

•994 

•0069 

2     1* 

8* 

51-85 

•360 

3| 

^ 

1-23 

•0085 

2     I* 

8* 

53-46 

•371 

4* 

f 

1-48 

•0103 

2     2* 

8f 

55-09 

•383 

-4f 

1 

1'77 

•0123 

2     2| 

8* 

56-74 

•394 

5* 

if 

2-07 

•0144 

2     3 

8| 

58-43 

•406 

5* 

If 

2-40 

•0167 

2     3| 

8| 

60-13 

•428 

«l 

It 

2-76 

•0192 

2     3l 

81 

61-86 

•430 

6* 

2 

3-14 

•0218 

2     4i 

9 

63-62 

•442 

6f 

2* 

3-55 

•0246 

2    4f 

gl 

65-40 

•455 

7 

2* 

3-98 

•0276 

2     5 

9* 

67-20 

•467 

7f 

2f 

4-43 

•0307 

2     5f 

Q3. 

69-03 

•480 

'71 

2* 

4-91 

•0341 

2     5f 

9* 

70-88 

•493 

8* 

21 

5-41 

•0376 

2     6* 

9f 

72-76 

•506 

8| 

2f 

5-94 

•0412 

2     6f 

9f 

74-66 

•519 

9 

2* 

6-49 

•0450 

2     7 

93 

76-59 

•532 

•1 

3 

7-07 

•0490 

2    7f 

10 

78-54 

•545 

10! 

1 

7'67 
829 

•0532 
•0576 

2     7f 
2     8* 

10* 
10* 

80-51 
82-52 

•559 
•573 

10f 

3f- 

8-95 

•0621 

2     8* 

iof 

84-54 

•587 

11 

3* 

9-62 

•0668 

2     9 

10* 

86-59 

•601 

111 

3f 

10-32 

•0716 

2     9f 

iof 

88-66 

•615 

111 

3f 

11-04 

•0766 

2     9f 

iof 

90-76 

•630 

12* 

81 

11-79 

•0818 

2  10* 

10* 

92-88 

•645 

1     0* 

4 

12-57 

•087 

2  10* 

11 

95-03 

•660 

1     1 

4* 

13-36 

•093 

2  101 

11* 

97-21 

•675 

If 

4* 

14-19 

•099 

2  11* 

11* 

99-40 

•690 

if 

4f 

15-03 

•105 

2  ll| 

llf 

101-62 

•705 

2* 

4* 

15-90 

•111 

3     0* 

11* 

103-87 

•720 

2* 

4| 

16-80 

•118 

3     0* 

llf 

106-14 

•736 

2* 

4f 

17-72 

•124 

3     0| 

lit 

108-43 

•752 

3* 

41 

18-66 

•130 

3     1* 

111 

110-75 

•768 

Sti- 
ff 

5 

19-63 

•136 

3     If 

12 

113-10 

•785 

4* 

5* 

20-63 

*14b 

3     2 

12* 

115-47 

•802 

1     4* 

5* 

21-65 

•150 

3     2* 

12* 

117-86 

•819 

43- 

Si 

22-69 

•157 

3     21 

12f 

120-28 

•836 

5i 

6* 

23-76 

•165 

o       04. 

12* 

122-72 

•853 

5f 

5f 

24-85 

•173 

3     3| 

12f 

125-19 

•870 

6 

5f 

25-97 

181 

3     4 

12f 

127-68 

•887 

6f 

5i 

27-11 

•189         1 

3     4| 

123- 

130-19 

•904 

692 


APPENDIX. 


TABLES  OF  THE  CIRCUMFERENCES  OF  CIECLES,  ETC.— (Continued.) 


Circumfer- 

Diameter 

Area 

Area 

Circumfer- 

Diameter 

Area 

Area 

ence  in  feet 

in 

in  square 

in  square 

ence  in  feet 

in  feet  and 

in  square 

in  square 

and  inches. 

inches. 

inches. 

feet. 

and  inches. 

inches. 

inches. 

feet. 

3     4f 

13 

132-73 

•922 

5     21 

20 

314-16 

2-182 

3     5i 

131 

135-30 

•939 

5     34 

201 

318-10 

2-209 

3     5f 

134 

137-89 

•956 

5     3f 

322-06 

2-237 

3     6 

13f 

140-50 

•974 

5     4 

20| 

326-05 

2-265 

3     6f 

143-14 

•992 

5     4f 

20£ 

330-06 

2-293 

3     6f 

13f 

145-80 

1-011 

5     41 

20f 

334-10 

2-321 

3     71 

13f 

148-49 

1-030 

6     5J 

20J 

338-16 

2-349 

3    7f 

ul 

151-20 

1-050 

5     o£ 

201 

342-25 

2-377 

3     8 

14 

153-94 

1-069 

5     6 

21 

346-36 

2-405 

3     8f 

14* 

156-70 

1-088 

5     6| 

21* 

350-50 

2-434 

3     8f 

144 

159-49 

1-107 

5     6| 

214 

354-66 

2-463 

3     91 

14f 

162-30 

1-126 

5     7^ 

21f 

358-84 

2-492 

3Q-L 
2 

14^ 

165-13 

1-146 

5     7i 

2l| 

363-05 

2-521 

3     91 

14f 

167-99 

1-166 

5     71 

21| 

367-28 

2-550 

3  104 
3  lOf 

141 

170-87 
173-78 

1-186 
1-206 

5     84 
5     8f 

211 

371-54 

375-83 

2-580 
2-610 

3  111 

15 

176-71 

1-227 

5     9J 

22 

380-13 

2-640 

3  Hi 

151 

179-67 

1-247 

5     9J 

221 

384-46 

2-670 

3  Hi 

154 

182-65 

1-267 

5     91 

224 

388-82 

2-700 

4     04 

15if 

185-66 

1-288 

5  10* 

22f 

393-20 

2-730 

4     Of 

15^ 

188-69 

1-309 

5   lOf 

224 

397-61 

2-761 

4     1 

15f 

191-75 

1-330 

5  11 

22f 

402-04 

2-792. 

4     1* 

15f 

194-83 

1-352 

5  Hi 

22* 

406-49 

2-823 

4     11 

151 

197-93 

1-374 

5  111 

221 

410-97 

2-854 

4     24 

16 

201-06 

1-396 

6     04 

23 

415-48 

2-885- 

4     2f 

161 

204-22 

1-418 

6     Of 

231 

420-00 

2-917 

4     3 

164 

207-39 

1-440 

6     1 

23^- 

424-56 

2-949 

4     3f 

16f 

210-60 

1-462 

6     If 

23| 

429-13 

2-981 

4     3| 

16^ 

213-82 

•484 

6     If 

23| 

433-74 

3-013 

4     44 

16f 

217-08 

•507 

6     24 

23f 

438-36 

3-045 

4    4| 

16f 

220-35 

•530 

6     2f 

28f 

443-01 

3-077 

4     5 

16! 

223-65 

•553 

6     3 

231 

447-69 

3-10& 

4     5f 

17 

226-98 

•576 

6     3| 

2     0 

452-39 

3-142. 

4     5f 

171 

230-33 

•599 

6     41 

2     04 

461-86 

3-207 

4     61 

233-70 

•622 

6     41 

2     0£ 

471-44 

3-273 

4     6i 

171 

237-10 

•645 

6     5< 

2     Of 

481-11 

3-341 

4     6l 

240-53 

•669 

6       6; 

2     1 

490-87 

3-408 

4    71 

171 

243-98 

•693 

6      7; 

2     14 

500-74 

3-477 

4    71 

17! 

247-45 

•718 

6     8- 

2    H 

510-71 

3-547 

4     8* 

171 

250-95 

•743 

6     8F 

2     If 

520-77 

3-617 

4     8^ 

18 

254-47 

•767 

6     9| 

2     2 

530-93 

3-687 

4     81 

181 

258-02 

•792 

6  10k 

2     24 

541-19 

3-758 

4     94 

184 

261-59 

•817 

6  114 

2     2J 

551-55 

3-830 

4     9f 
4  101 

18* 
18* 

265-18 
268-80 

•842 
•868 

7     0 
7     01 

2     2f 
2     3 

562-00 
572-56 

3-904 
3-976 

4  10i 

18f 

272-45 

•893 

7     If 

2     34 

583-21 

4-050 

4  101 
4  114 

1?! 

276-12 
279-81 

•918 
•943 

7     2f 

2     3* 

2     3f 

59396 
604-81 

4-124 
4-200 

4  Hf 

19 

283-53 

1-969 

7    31 

2     4 

615-75 

4-276 

5     0 

19i 

287-27 

1-995 

7    4f 

2     44 

626-80 

4-352 

6     Of 

194 

291-04 

2-021 

7    5* 

2     4* 

637-94 

4-430 

5     01 

294-83 

2-047 

7     64 

2     41 

649-18 

4-508 

5     14 

19| 

298-65 

2-074 

7    7 

2     5 

660-52 

4-586 

5     If 

19s 

302-49 

2-101 

7    7i 

2     54 

671-96 

4-666 

5     2 

19f 

306-36 

2-128 

7     83- 

2     5| 

683-49 

4-747 

6     2| 

310-25 

2-155 

7     9£ 

2     5f 

695-13 

4-827 

APPENDIX. 


693 


TABLES   OF  THE  CIECUMFEEENCES  OF  CIRCLES,  ETC.— (Continued.) 


Circumfer-    '     Diameter 

Area 

Area 

Circumfer- 

Diameter 

Area 

Area 

«nce  in  feet 
and  inches. 

in  feet  and 
inches. 

in  square 
inches. 

in  square 
feet. 

ence  in  teet 
and  inches. 

in  feet  and 
inches. 

in  square 
inches. 

in  square 
feet. 

7  10i 

2     6 

706-86 

4-908 

11     6J 

3     8 

1520-5 

10-56 

7   11 

2     6J- 

718-69 

4-990 

11      7 

3     8± 

1537-9 

10-68 

7  llf 

2     64 

730-62 

5-073 

11     7f 

3     8* 

1555-3 

10-80 

8     0| 

2     6f 

742-64 

6-157 

11     8| 

3     8f 

1572-8 

10-92 

8     1| 

2     7 

754-77 

5-241 

11     9| 

3     9 

1590-4 

11-04 

8     2^r 

2     7£ 

766-99 

5-326 

11   10£ 

3     9i 

1608-1 

11-17 

8     2£ 

2     7| 

779-31 

5-411 

11   103            3     9| 

1626-0 

11-29 

8     3f 

2     7f 

791-73 

5-498 

11   llf           3     9f 

1643-9 

11-41 

8     4| 

2     8 

804-25 

5-585 

12     0| 

3  10 

1661-9 

11-54 

8     5| 

2     8i 

816-86 

5-673 

12     lj 

3  101 

1680-0 

11-67 

8     6j 

2     8| 

829-58 

5-761 

12     2 

3  10! 

1698-2 

11-79 

8     6£ 

2     8f 

842-39 

5-849 

12     2| 

3  lOf 

1716-5 

11-92 

8     71 

2     9 

855-30 

5-939 

12     8f 

3  11 

1734-9 

12-05 

8     8fr 

2     9^ 

868-31 

6-029 

12     4| 

3  iii 

1753-4 

12-18 

8     9i 

2     9| 

881-41 

6-120 

12     5* 

3  11J 

1772-0 

12-30 

8  10 

2     9f 

894-62 

6-212 

12     6 

3  llf 

1790-8 

12-43 

8  lOf 

2  10 

907-92 

6-305 

12     6f 

4     0 

1809-6 

12-57 

8  11| 

2  10  J 

921-32 

6-398 

12     7| 

4     0| 

1828-5 

12-70 

9     Of- 

2  10J 

934-82 

6-491 

12     8| 

4     0| 

1847-4 

12-83 

9     1| 

2  101 

948-42 

6-586 

12     9| 

4     Of 

1866-5 

12-96 

9     13 

2  11 

962-11 

6-681 

12     93 

4     1 

1885-7 

13-09 

9     2f 

2  11J 

975-91 

6-777 

12  lOf 

4     11 

1905-0 

13-23 

9     3£ 

2  H| 

989-80 

6-874 

12  11| 

4    l! 

1924-4 

13-36 

9     4-J- 

2  llf 

1003-8 

6-970 

13     0| 

4     If 

1943-9 

13-50 

9     5 

3     0 

1017-9 

7-069 

13     1 

4     2 

1963-5 

13-63 

9     5J 

3     Oi 

1032-1 

7-167 

13     13 

4     2i 

1983-2 

13-77 

9     6| 

3     0| 

1046-3 

7-266 

13     2f 

4     2| 

2003-0 

13-91 

9     7i 

3     Of 

1060-7 

7-366 

13     3f 

4     2f 

2022-8 

14-05 

9     8i 

3     1 

1075-2 

7-466 

13     4J 

4     3 

2042-8 

14-19 

9     9 

3     1J 

1089-8 

7-567 

13     5 

4     3£ 

2062-9 

14-32 

9     9| 

3     1J 

1104-5 

7-669 

13     5f 

4     3! 

2083-1 

14-46 

9  lOf 

3     If 

1119-2 

7-772 

13     6| 

4     3f 

210S-3 

14-61 

9  llf 

3     2 

1134-1 

7-876 

13     7f 

4     4 

2123-7 

14-75 

10    0| 

3     2| 

1149-1 

7-979 

13     8i 

4     4J 

2144-2 

14-89 

10     OI- 

3     2| 

1164-2 

8-085 

13     8| 

4     4| 

2164-7 

15-03 

IO     If 

3     2| 

1179-3 

8-189 

13     9f 

4     4f 

2185-4 

15-18 

10     2| 

3     3 

1194-6 

8-295 

13  10| 

4     5 

2206-2 

15-32 

10     3J 

3O  1 
"i 

1209-9 

8-403 

13  ll| 

4     5£ 

2227-0 

15-46 

10     4 

3     3£  • 

1225-4 

8-509 

14     0 

4     5| 

2248-0 

15-61' 

10     43 

3     3f 

1241-0 

8-617 

14     03 

4     5| 

2269-1 

15-76 

10     5| 

3     4 

1256-6 

8-727 

14     If 

4     6 

2290-2 

15-90 

10     6| 

3     4\ 

1272-4 

8-836 

14     2f 

4     6£ 

2311-5 

16-05 

10     7i 

3     4i 

1288-2 

8-946 

14     3i 

4     6| 

2332-8 

16-20 

10     8 

3     4f 

1304-2 

9-056 

14     4 

4     6f 

2354-3 

16-35 

10     81 

3     5 

1320-2 

9-169 

14     4f 

4    7 

2375-8 

16-50 

10     Qk 

3     5J- 

1336-4 

9-211 

14     5£ 

4    7i 

2397-5 

16-65 

10  10J 

3     5£ 

1352-6 

9-394 

14     6| 

4    7i 

2419-2 

16-80 

10  11| 

3     5f 

1369-0 

9-506 

14     7| 

4    7f 

2441-1 

16-95 

10  113 

3     6 

1385-4 

9-62 

.    14    7£ 

4     8 

2463-0 

17-10 

11     Of 

3     6i 

1402-0 

9-73 

14     83 

4     8| 

2485-0 

17-26 

11    i| 

3     6| 

1418-6 

9-84 

14     9! 

4     8| 

2507-2 

17-41 

11     2± 

3     6f 

1435-4 

9-96 

14  10i 

4     8f 

2529-4 

17-56 

11     3 

3     7 

1452-2 

10-08 

14  11 

4     9 

2551-8 

17-72 

11     33 

3     7i 

1469-1 

10-20 

14  H| 

4     9| 

2574-2 

17-88 

11     4| 

3     7| 

1486-2 

10-32 

15     Of 

4     9| 

2596-7 

18-03 

11     5| 

3     7f 

1503-3 

10-44 

15     If 

4     9f 

2619-3 

18-19 

694 


APPENDIX. 


TABLES   OF  THE  CIRCUMFERENCES   OF  CIRCLES,  ETC.— (Continued.} 


Circumfer- 

Diameter 

Area 

Area 

Circumfer- 

Diameter 

Area 

Area 

ence  in  feet 
and  inches. 

in  feet  and 
inches. 

in  square 
inches. 

in  square 
feet. 

|  ence  in  feet 
i  and  inches. 

in  feet  and 
inches. 

in  square 
inches. 

in  square- 
feet. 

15  2t 

4  10 

2642-1 

18-35 

18  10i 

6  0 

4071-5 

28-27 

15  3 

4  10t 

2664-9 

18-51 

18  10* 

6  01 

4099-8 

28-47 

15  3f 

4  10£ 

2687-8 

18-66 

18  llf 

6  Oi 

4128-2 

28-67 

15  4i 

4  10$ 

2710-8 

18-82 

19  0£ 

6  Of 

4156-8 

28-87 

15  5{ 

4  11 

2734-0 

18-98 

19  l| 

6  1 

4185-4 

29-07- 

15  61 

4  iii 

2757-2 

19-15 

19  2j 

6  li 

4214-1 

29-27 

15  6| 

4  iii 

2780-5 

19-31 

19  23 

6  li 

4242-9 

29-47 

15  7f 

4  llf 

2803-9 

19-47 

19  3f 

6  1| 

4271-8 

29-67' 

15  8i 

5  0 

2827-4 

19-63 

19  4i 

6  2 

4300-8 

29-87- 

15  91 

5  Oi 

2851-0 

19-80 

19  51 

6  2i 

4329-9 

30-07 

15  10 

5  Oi 

2874-8 

19-96 

19  6 

6  2i 

4359-2 

30-27 

15  lOf 

5  Of 

2898-6  j   20-13 

19  6f 

6  21 

4388-5 

30-47 

15  llf 

5  1 

2922-5 

20-29 

19  7i 

6  3 

4417-9 

30-68- 

16  Of 

5  It 

2946-5 

20-46 

19  8f 

6  31 

4447-4 

30-88-. 

16  U 

5  1£ 

2970-6 

20-63 

19  9i 

6  3$ 

4477-0 

31-0& 

16  2 

5  If 

2994-8 

20-80 

19  9| 

6  3f 

4506-7 

31-3O- 

16  2f 

5  2 

3019-1 

20-96 

19  lOf 

6  4 

4536-5 

31-50- 

16  3i 

6  2i 

3043-5 

21-13 

19  Hi 

6  4i 

4566-4 

31-71 

16  4t 

5  2i 

3068-0 

21-30 

20  01 

6  4i 

4596-3 

31-92 

16  5£ 

5  2| 

3092-6 

21-48 

20  li 

6  4f 

4626-4 

32-13- 

16  Si 

5  3 

3117-2 

21-65 

20  li 

6  5 

4656-6 

32-34 

16  6f 

5  3i 

3142-0 

21-82 

20  2f 

6  5i 

4686-9 

32-55 

16  7* 

5  3| 

3166-9 

21-99 

20  3i 

6  ef 

4717-3 

3276. 

16  8} 

5  3f 

3191-9 

22-17 

20  41 

6  6f 

4747-8 

32-97- 

16  9 

5  4 

3217-0 

22-34 

20  6 

6  6 

4778-3 

33-18 

16  9f 

5  4t 

3242-2 

22-51 

20  5| 

6  61 

4809-0 

33-40" 

16  10| 

5  4£ 

3267-5 

22-69 

20  6£ 

6  6J 

4839-8 

33-61 

16  ll| 

5  4f 

3292-8 

22-87 

20  7f 

6  6| 

4S70-7 

33-82^: 

17  Oi 

5  5 

3318-3 

23-04 

20  8£ 

6  7 

4901-6 

34-04 

17  1 

5  Si 

3343-9 

23-22 

20  8| 

6  71 

4932-7 

34-25 

17  If 

5  5£ 

3369-6 

23-40 

20  9f 

6  7^ 

4963-9 

34-47 

17  2J 

5  5f 

3395-3 

23-58 

20  10} 

6  7f 

4995-1 

34-69- 

17  3| 

5  6 

3421-2 

23-76 

20  111 

6  8 

5026-5 

34-91 

17  4| 

6  6£ 

3447-2 

23-94 

21  0£ 

6  81 

6058-0 

35-12' 

17  4| 

5  6£ 

3473-2 

24-12 

21  0| 

C   oiy 

5089-5 

35-34 

17  5£ 

5  6f 

3499-4 

24-30 

21  If 

6  8f 

5121-2 

35-56 

17  6i 

5  7 

3525-1 

24-48 

21  2f 

6  9 

5153-0 

35-78. 

17  7i 

5  71 

3552-0 

24-67 

21  31 

6  91 

6184-8 

36-01 

17  8 

5  7i 

3578-5 

24-85 

21  4 

6  9i 

5216-8 

36-23 

17  8f 

6  7| 

3605-0 

25-03 

21  4| 

6  9j 

5248-8 

36-45 

17  9f 

5  8 

3631-7 

25-22 

21  5| 

6  10 

5281-0 

36-67' 

17  lOf 

5  8i 

3658-4 

25-40 

21  6f 

6  101 

5313-2 

36-89- 

17  lit 

5  8i 

3685-3 

25-59 

21  7j 

6  10| 

5345-6 

37-12- 

17  11| 

5  8| 

3712-2 

25-78 

21  7| 

6  lOf 

6378-0 

37-35 

18  Of 

5  9 

3739-3 

25-96 

21  8| 

6  11 

5410-6 

37-57 

18  li 

5  9i 

3766-4 

26-15 

21  9£ 

6  lit 

5443-2 

87-80* 

18  2i 

6  9i 

3793-7 

26-34 

21  101 

6  11£ 

5476-0 

38-03- 

18  3i 

5  9f 

3821-0 

26-53 

21  111 

6  llf 

5508-8 

38-2& 

18  3£ 

5  10 

3848-5 

26-72 

21  llf 

7  0 

5541-7 

38-48 

18  4f 

5  10i 

3876-0 

26-92 

22  Of 

7  01 

5574-8 

38-71 

18  Si 

5  10J- 

3903-6 

27-11 

22  If 

7  Oi 

5607-9 

38-94 

18  6i 

6  lOf 

3931-4 

27-30 

22  21 

7  9| 

6641-1 

39-17 

18  7 

6  11 

3959-2 

27-49 

22  3 

7  1 

5674-5 

39-41 

18  7| 

5  Hi 

3987-1 

27-69 

22  3* 

7  It 

5707-9 

39-64 

18  8| 

5  Hi 

4015-2 

27-88 

22  4i 

7  1? 

6741-4 

39-87 

18  9| 

5  llf 

4043-3 

28-08 

22  Si- 

7  If 

5775-0 

40'10> 

APPENDIX. 

TABLES  OF  THE  CIRCUMFERENCES  OF  CIRCLES,   ETC.— (Continued.) 


695 


Circumfer- 

Diameter 

Area 

Area 

Circumfer- 

Diameter 

Area 

Area 

ence  in  feet 
and  inches. 

in  feet  and 
inches. 

in  square 
inches. 

in  square 
feet. 

ence  in  feet 
and  inches. 

in  feet  and 
inches. 

in  square 
inches. 

in  square 
feet. 

22  6J 

7  2 

5808-8 

40-34 

26  21 

8  4 

7853-9 

54-54 

22  61 

7  2} 

5842-6 

40-57 

26  6J 

8  5 

8011-9 

55-64 

22  71- 

I    £A 

5876-5 

40-80 

26  8| 

8  6 

8171-3 

56-75 

22  8£ 

7  2| 

5910-5 

41-04 

26  ll£ 

8  7 

8332-3 

57-86 

22  9± 

7  3 

5944-6 

41-28 

27  2f 

8  8 

8494-9 

58-99 

22  10£ 

7  3± 

5978-9 

41-52 

27  5t 

8  9 

8659-0 

60-13 

22  101 

6013-2 

41-76 

27  9 

8  10 

8824-7 

61-28 

22  11| 

7  3f 

6047-6 

42-00 

28  01 

8  11 

8892-0 

62-44 

23  Of 

7  4 

6082-1 

42-24 

28  3^ 

9 

9160-9 

63-62 

23  U 

7  4i 

6116-7 

42-48 

28  6| 

9  1 

9331-3 

64-80 

23  2 

7  44 

6151-4 

42-72 

28  9} 

9  2 

9503-3 

66-00 

23  2| 

7  4| 

6186-2 

42-96 

29  Of 

9  3 

9676-9 

67-20 

23  3f 

7  5 

6221-1 

43-20 

29  3f 

9  4 

9852-1 

68-42 

23  4f 

7  6t 

6256-1 

43-44 

29  7 

9  5 

10028-8 

69-64 

23  5£ 

7  6* 

6291-2 

43-68 

29  10J 

9  6 

10207-1 

70-88 

23  6 

7  51 

6326-4 

43-93 

30  1£ 

9  7 

10386-9 

72-13 

23  6| 

7  6 

6361-7 

44-18 

30  4f 

9  8 

10568-3 

73-39 

23  7£ 

7  6i 

6397-1 

44-43 

30  74 

9  9 

10751-3 

74-66 

23  8£ 

7  6J 

6432-6 

44-67 

30  lOf 

9  10 

10935-9 

75-94 

23  9| 

7  6| 

6468-2 

44-92 

31  If 

9  11 

11122-0 

77-24 

23  91 

7  7 

6503-8 

45-17 

23  10£ 

7  7i 

6539-6 

45-41 

31  5 

10 

11309-8 

78-54 

23  llf 

7  7£ 

6575-5 

45-66 

31  81 

10  1 

11499-0 

79-85 

24  0^ 

7  7f 

6611-5 

45-91 

31  ll| 

10  2 

11689-9 

81-18 

32  2f 

10  3 

11882-3 

82-52 

24  1 

7  8 

6647-6 

46-16 

32  5| 

10  4 

12076-3 

83-86 

24  H 

7  8J 

6683-8 

46-42 

32  8f 

10  5 

12271-9 

85-22 

24  2* 

7  8v 

6720-0 

46-67 

32  llf 

10  6 

12469-0 

86-59 

24  3j 

7  81 

6756-4 

46-92 

33  21 

10  7 

12667-7 

87-97 

24  41 

7  9 

6792-9 

47-17 

33  6| 

10  8 

12868-0 

89-36 

24  41 

7  9i 

•6829-4 

47-43 

33  9± 

10  9 

13069-8 

90-76 

24  51 

7  9J 

6866-1 

47-68 

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10  10 

13273-3 

92-17 

24  6£ 

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6902-9 

47-94 

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10  11 

13478-2     93-60 

24  7i 

7  10 

6939-7 

48-19 

34  6$ 

11 

13684-8 

95-03 

24  8 

7  101 

6976-7 

48-45 

34  9f 

11  1 

13892-9 

96-48 

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7  lOfc 

7013-8 

48-71 

35  0^ 

11  2 

14142-6     97-93 

24  9| 

7  10| 

7050-9 

48-96 

35  41 

11  3 

14313-9     99-40 

24  10| 

7  11 

7088-2 

49-22 

35  7J 

11  4 

14526-8 

100-88 

24  llf 

7  11J 

7125-5 

49-48 

35  lOf 

11  5 

14741-2 

102-37 

25  0 

7  ll| 

7163-0 

49-74 

36  l| 

11  6 

14957-2 

103-87 

25  Of 

7  llf 

7200-5 

50-00 

36  4f 

11  7 

15174-7 

105-38 

36  7f 

11  8 

15393-8 

106-90 

25  1£ 

8  0 

7238-2 

50-26 

36  101 

11  9 

15614-5 

108-43 

25  2f 

8  OJ 

7275-9 

50-53 

37  2 

11  10 

15836-8 

109-98 

25  3J 

8  o| 

7313-8 

50-79 

37  5J 

11  11 

16060-6 

111-53 

25  31 

8  Of 

7351-7 

51-05 

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7389-8 

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7427-9 

51-58 

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12  1 

16513-0 

114-67 

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7466-2 

51-85 

38  2f 

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16741-6 

116-26 

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8  if 

7504-5 

52-11 

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12  3 

16971-7 

117-86 

38  81 

12  4 

17203-4 

119-47 

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12  5 

17436-7 

121-09 

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8  21 

7581-5 

52-65 

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124-36 

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53-19 

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12  8 

18145-9 

126-01 

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127-68 

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7736-6 

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54-00 

40  61 

12  11 

18869-2 

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8  3f 

7814-7 

54-27 

696 


APPENDIX. 


TABLE  OF  SQUARES,   CUBES,   SQUARE  AND  CUBE  ROOTS  OF    NUMBERS. 


Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

1         1 

1 

I'OOO 

1-000 

4096 

262144 

64 

8-000 

4-000 

4        8 

2 

1-414 

1-259 

4225 

274625 

65 

8-062 

4-020 

9       27 

3 

1-732 

1-442 

4356 

287496 

66 

8-124 

4-041 

16       64 

4 

2-000 

1-587 

4489 

300763 

67 

8-185 

4-061 

25 

125 

5 

2-236 

1-709 

4624 

314432 

68 

8-246 

4-081 

36 

216 

6 

2-449 

1-817 

4761 

328509 

69 

8-306 

4-101 

49 

343 

7 

2-645 

1-912 

4900 

343000 

70 

8-366 

4-121 

64      612 

8 

2-828 

2-000 

5041 

357911 

71 

8-426 

4-140 

81      729 

9 

3-000 

2-080 

5184 

373248 

72 

8-485 

4-160 

100     1000 

10 

3-162 

2-154 

5329 

389017 

73 

8-544 

4-179 

121 

1331 

11 

3-316 

2-223 

5476 

405224 

74 

8-602 

4-198 

144 

1728 

12 

3-464 

2-289 

5625 

421875 

75 

8-660 

4-217 

169 

2197 

13 

3-605 

2-351 

5776 

438976 

76 

8-717 

4-235 

196 

2744 

14 

3-741 

2-410 

5929 

456533 

77 

8-774 

4-254 

225     3375 

15 

3-872 

2-466 

6084 

474552 

78 

8-831 

4-272 

256     4096 

16 

4-000 

2-519 

6241 

493039 

79 

8-888 

4-290 

289  :    4913 

17 

4-123 

2-571 

6400 

512000 

80 

8-944 

4-308 

324 

6832 

18 

4-242 

2-620 

6561 

531441 

81 

9-000 

4-326 

361 

6859 

19 

4-358 

2-668 

6724 

551368 

82 

9-055 

4-344 

400 

8000 

20 

4-472 

2-714 

6889 

571787 

83 

9-110 

4-362 

441 

9261 

21 

4-582 

2-758 

7056 

592704 

84 

9-165 

4-379 

484 

10648 

22 

4-690 

2-802 

7225 

614125 

85 

9-219 

4-396 

629 

12167 

23 

4-795 

2-843 

7396 

636056 

86 

9-273 

4-414 

676 

13824 

24 

4-898 

2-884 

7569 

658503 

87 

9-327 

4-431 

625 

15625 

25 

5-000 

2-924 

7744 

681472 

88 

9-380 

4-447 

676 

17576 

26 

5-099 

2-962 

7921 

704969 

89 

9-433 

4-464 

729 

19683 

27 

5-196 

3-000 

8100 

729000 

90 

9-486 

4-481 

784 

21952 

28 

5-291 

3-036 

8281 

753571 

91 

9-539 

4497 

841 

24389 

29 

5-385 

3-072 

8464 

778688 

92 

9-591 

4-514 

900 

27000 

30 

5-477 

3-107 

8649 

804357 

93 

9-643 

4-530 

961 

29791 

31 

5-567 

3-141 

8836 

830584 

94 

9-695 

4-546 

1024 

32768 

32 

5-656 

3-174 

9025 

857374 

95 

9-746 

4-562 

1089 

35937 

33 

5-744 

3-207 

9216 

884736 

96 

9-797 

4-578 

1156 

39304 

34 

5-830 

3-239 

9409 

912673 

97 

9-848 

4-594 

1225 

42875 

35 

5-916 

3-271 

9604 

941192 

98 

9-899 

4-610 

1296 

46656 

36 

6-000 

3-301 

9801 

970299 

99 

9-949 

4-626 

1369 

60653 

37 

6-082 

3-332 

10000 

1000000 

100 

10-000 

4-641 

1444 

54872 

38 

6-164 

3-361 

10201 

1030301 

101 

10-049 

4-657 

1521 

59319 

39 

6-244 

3-391 

10404 

1061208 

102 

10-099 

4-672 

1600 

64000 

40 

6-324 

3-419 

10609 

1092727 

103 

10-148 

4-687 

1681 

68921 

41 

6-403 

3-448 

10816 

1124864 

104 

10-198 

4-702 

1764 

74088 

42 

6-480 

3-476 

11025 

1157625 

105 

10-246 

4-717 

1849 

79507 

43 

6-557 

3-503 

11236 

1191016 

106 

10-295 

4-732 

1936 

85184 

44 

6-633 

3-530 

11449 

1225043 

107 

10-344 

4-747 

2025 

91125 

45 

6-708 

3-556 

11664 

1259712 

108 

10-392 

4-762 

2116    97336 

46 

6-782 

3-583 

11881 

1295029 

109 

10-440 

4-776 

2209 

103823 

47 

6-855 

.  3-608 

12100 

1331000 

110 

10-488 

4-791 

2304 

110592 

48 

6-928 

3-634 

12321 

1367631 

111 

10-535 

4-805 

2401    117649    49 

7*000 

3-659 

12544 

1404928 

112 

10-583 

4-820 

2500 

125000    50 

7-071 

3-684 

12769 

1442897 

113 

10-630 

4-834 

2601 

132651     51 

7-141 

3-708 

12996 

1481544 

114 

10-677 

4-848 

2704 

140608 

52 

7-211 

3-732 

13225 

1520875 

115 

10-723 

4-862 

2809 

148877 

53 

7-280 

3-756 

13456 

1560896 

116 

10-770 

4-876 

2916 

157464 

54 

7-348 

3-779 

13689 

1601613 

117 

10-816 

4-890 

3025 

166375 

55 

7-416 

3-802 

13924 

1643032 

118 

10-862 

4-904 

3136 

175616 

56 

7-4S3 

3-825 

14161 

1685159 

119 

10-908 

4-918 

3249 

185193 

57 

7-549 

3-848 

14400 

1728000 

120 

10-954 

4-932 

3364 

195112 

58 

7-615 

3-870 

14641 

1771561 

121 

11-000 

4-946 

3481 

205379 

59 

7-681 

3-892 

14834 

1815848 

122 

11-045 

4-959 

3600 

216000 

60 

7-745 

3-914 

15129 

1860867 

123 

11-090 

4-973 

3721 

226981 

61 

7-810 

3-930 

15376 

1906624 

124 

11-135 

4-986 

3844 

238328 

62 

7-874 

3-957 

15625 

1953125 

125 

11-180 

6-000 

3969 

250047 

63 

7-937 

3-979 

15876 

2000376 

126 

11-224 

5-013 

APPENDIX. 


697 


TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS   OF  NUMBERS— ( Continued). 


Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

16129 

2048383 

127 

11-269 

5-026 

36100 

6859000   190 

13-784 

5-748 

16384 

2097152 

128 

11-313 

5-039 

36481 

6967871    191 

13-820 

5-758 

16641 

2146689 

129 

11-357 

5-052 

36864 

7077888   192 

13-856 

5-768 

16900 

2197000 

130 

11-401 

5-065 

37249 

7189517    193 

13-892 

5-778 

17161 

2248091 

131 

11-445 

5-078 

37636 

7301384 

194 

13-928 

5-788 

17424 

2299968 

132 

11-489 

5-091 

38025 

7414875 

195 

13-964 

6-798 

17689 

2352637 

133 

11-532 

5-104 

38416 

7529536 

196 

14-000 

5-808 

17956 

2406104 

134 

11-575 

5-117 

38809 

7645373 

197 

14-035 

5-818 

18225 

2460375 

135 

11-618 

5-129 

39204 

7762392 

198 

14-071 

B'828 

18496 

2515456 

136 

11-661 

5-142 

39601 

7880599 

199 

14-106 

5-838 

18769 

2571353 

137 

11-704 

5-155 

40000 

8000000 

200 

14-142 

5-848 

19044 

2628072 

138 

11-747 

5-167 

40401 

8120601 

201 

14-177 

6-857 

19321 

2685619 

139 

11-789 

5-180 

40804 

8242408 

202 

14-212 

5-867 

19600 

2744000 

140 

11-832 

5-192 

41209 

8365427 

203 

14-247 

5-877 

19881 

2803221 

141 

11-874 

5-204 

41616 

8489664 

204 

14-282 

6-886 

20164 

2863288 

142 

11-916 

5-217 

42025 

8615125 

205 

14-317 

5-896 

20449 

2924207 

143 

11-958 

5-229 

42436 

8741816 

206 

14-352 

5-905 

20736 

2985984 

144 

12-000 

5-241 

42849 

8869743 

207 

14-387 

5-915 

21025 

3048625 

145 

12-041 

5-253 

43264 

8998912 

208 

14-422 

5-924 

21316 

3112136 

146 

12-083 

5-265 

43681 

9129329 

209 

14-456 

5-934 

21609 

3176523 

147 

12-124 

5-277 

44100 

9261000 

210 

14-491 

5-943 

21904 

3241792 

148 

12-165 

5-289 

44521 

9393931 

211 

14-525 

5-953 

22201 

3307949 

149 

12-206 

5-301 

44944 

9528128 

212 

14-560 

5-962 

22500 

3375000 

150 

12-247 

5-313 

45369 

9663597 

213 

14-594 

5-972 

22801 

3442951 

151 

12-288 

5-325 

45796 

9800344 

214 

14-628 

5-981 

23104 

3511008 

152 

12-328 

5-336 

46225 

9938375 

215 

14-662 

5-990 

23409 

3581577 

153 

12-369 

5-348 

46656 

10077696 

216 

14-696 

6-000 

23716 

3652264 

154 

12-409 

5-360 

47089 

10218312 

217 

14-730 

6-009 

24025 

3723875 

155 

12-449 

5-371 

47524 

10360232 

218 

14-764 

6-018 

24336 

3796416 

156 

12-489 

5-383 

47961 

10503459 

219 

14-798 

6-027 

24649 

3869893 

157 

12-529 

5-394 

48400 

10648000 

220 

14-832 

6-036 

24964 

3944312 

158 

12-569 

5-406 

48841 

10793861 

221 

14-866 

6-045 

25281 

4019679 

159 

12-609 

5-417 

49284 

10941048 

222 

14-899 

6-055 

25600 

4096000 

160   12-649 

5-428  ' 

49729 

11089567 

223 

14-933 

6-064 

25921 

4173281 

161 

12-688 

5-440 

50176 

11239424 

224 

14-966 

6-073 

26244 

4251528 

162 

12-727 

5-451 

50625 

11390625 

225 

15-000 

6-082 

26569 

4330747 

163 

12-767    5-462 

51076 

11543176 

226 

15-033 

6-099 

26896 

4410944 

164 

12-806 

5-473 

51529 

11697083 

227 

15-066 

6-100 

27225 

4492125 

165 

12-845 

5-484 

51984 

11852352 

228 

15-099 

6-109 

27556 

4574296 

166 

12-884    5-495 

52441 

12008989 

229 

15-132 

6-118 

27889 

4657463 

167 

12-922 

5'506 

52900 

12167000 

230 

15-165 

6-126 

28224 

4741632 

168 

12-961 

5-517 

53361 

12326391 

231 

15-198 

6-135 

28561 

4826809 

169 

13-000 

5-528 

53824 

12487168 

232 

15-231 

6-144 

28900 

4913000 

170 

13-938 

5-539 

54289 

12649337 

233 

15-264 

6-153 

29241 

5000211 

171 

13-076 

5-550 

54756 

12812904 

234 

15-297 

6-162 

29584 

5088448 

172 

13-114 

5-561 

55225 

12977875 

235 

15-329 

6-171 

29929 

6177717 

173 

13-152 

5-572 

55696 

13144256 

236 

15-362 

6-179 

30276 

5268024 

174 

13-190 

5-582 

56169 

13312053 

237 

15-394 

6-188 

30625 

5359375 

175 

13-228 

5-593 

56644 

13481272 

238 

15-427 

6-197 

30976 

5451776 

176 

13-266 

5-604 

57121 

13651919 

239 

15-459 

6-205 

31329 

5545233 

177 

13-304 

5-614 

57600 

13824000 

240 

15-491 

6-214 

31684 

5639752 

178 

13341 

5-625 

58081   13997521 

241 

15-524 

6-223 

32041 

5735339 

179 

13-379 

5-635 

58564 

14172488 

242 

15-556 

6-231 

32400 

58S2000 

180 

13-416 

5-646 

59049 

14348907 

243 

15-588 

6-240 

32761 

5929741 

181 

13-453 

5-656 

59536 

14526784 

244 

15-620 

6-248 

33124 

6028568 

182 

13-490 

5-667 

60025 

14706125 

245 

15-652 

6-257 

33489 

6128487 

183 

13-527 

5-677 

60516 

14886936 

246 

15-684 

6-265 

33856 

6229504 

184 

13-664 

5-687 

61009 

15069223 

247 

15-716 

6-274 

34225 

6331625 

185 

13-601 

5-698 

61504 

15252992 

248 

15-748 

6-282 

34596 

6434856 

186 

13-638 

5-708 

62001 

15438249 

249 

15-779 

6-291 

34969 

6539203 

187 

13-674 

5-718 

62500 

15625000 

250 

15-811 

6-299 

35344 

6644672 

188 

13-711 

5-728 

63001 

15813251 

251 

15-842 

6-307 

35721 

6751269 

189 

13-747   5-738 

63504 

16003008 

252 

15-874 

6-316 

698 


APPENDIX. 


TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS  OF  NUMBERS— (Continued}, 


Squares.    Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

64009   16194277 

253 

15-905 

6-324 

99856 

31554496    316 

17-776 

6-811 

64516 

16387064 

254   15-937 

6-333 

100489 

31855013    317 

17-804 

6-818 

65025 

16581375 

255   15-968 

6-341 

101124 

32157432    318 

17-832 

6-825 

65536 

16777216 

256  !  16-000 

6-349 

101761 

32461759    319 

17-860 

6-832 

66049 

16974593 

257   16-031 

6-357 

102400 

32768000   320 

17-888 

6-839 

66564 

17173512 

258   16-062 

6-366 

103041 

33076161    321 

17-916 

6-847 

67081 

17373979 

259 

16093 

6-374 

103684 

33386248    322 

17-944 

6-854 

67600 

17576000 

260 

16-124 

6-382 

104329 

33698267    323 

17-972 

6-861 

68121 

17779581 

261 

16-155 

6-390 

104976 

34012224 

324 

18-000 

6-868 

68644 

17984728 

262 

16-186 

6-398 

105625 

34328125 

325 

18-027 

6-875 

69169 

18191447 

263 

16-217 

6-406 

106276 

34645976    326 

18-055 

6-882 

69696 

18399744 

264 

16-248 

6-415 

106929 

34965783    327 

18-083 

6-889 

70225 

18609625 

265 

16-278 

6-423 

107584 

35287552 

328 

18-110 

6-896 

70756 

18821096 

266 

16-309 

6-431 

108241 

35611289 

329 

18-138 

6-903 

71289 

19034163 

267 

16-340 

6-439 

108900 

35937000 

330 

18-165 

6-910 

71824 

19248832 

268 

16-370 

6-447 

109561 

36264691 

331 

18'  193 

6-917 

72361 

19465109 

269 

16-401 

6-455 

110224 

36594368 

332 

18-220 

6-924 

72900 

19683000 

270 

16-431 

6-463 

110889 

36926037 

333 

18-248 

6-931 

73441 

19902511 

271 

16-462 

6-471 

111556 

37259704 

334 

18-275 

6-938 

73984 

20123643 

272 

16-492 

6-479 

112225 

37595375 

335 

18-303 

6-945 

74529 

20346417 

273 

16-522 

6-487 

112896 

37933056 

336 

18-330 

6-952 

75076 

20570824 

274 

16-552 

6-495 

113569 

38272753 

337 

18-357 

6-958 

75625 

20796875 

275 

16-583 

6-502 

114244 

38614472 

388 

18-384 

6-965 

76176 

21024576 

276 

16-613 

6-510 

114921 

38958219 

339 

18-411 

6-972 

76729 

21253933 

277 

16-643 

6-518 

115600 

39304000 

340 

18-439 

6-979 

77284 

21484952 

278 

16-678 

6-526 

116281 

89651821 

341 

18-466 

6-986 

77841 

21717639 

279 

16-703 

6-534 

116964 

40001688 

342 

18-493 

6-993 

78400 

21952000 

280 

16-733 

6-542 

117649 

40353607 

343 

18-520 

7-000- 

78961 

22188041 

281 

16-763 

6-549 

118336 

40707584 

344 

18-547 

7-006 

79524 

22425768 

282 

16-792 

6-557 

119025 

41063625 

345 

18-574 

7-013 

80089 

22665187 

283 

16-822 

6-565 

119716 

41421736 

346 

18-601 

7-020 

80656 

22906304 

284 

16-852 

6-573 

120409 

41781923 

347 

18-627 

7-027 

81225 

23149125 

285 

16-881 

6-580 

121104 

42144192 

348 

18-654 

7-033 

81796 

23393656 

286 

16-911 

6-588 

121801 

42508549 

349 

18-681 

7-040 

82369 

23639903 

287 

16-941 

6-596 

122500 

42875000 

350 

18-708 

7-047 

82944 

23887872 

288 

16-970 

6-603 

123201 

43243551 

351 

18-734 

7-054 

83521 

24137569 

289 

17-000 

6-611 

123904 

43614208 

352 

18-761 

7-060 

84100 

24389000 

290 

17-029 

6-619 

124609 

43986977 

353 

18-788 

7-06Y 

84681 

24642171 

291 

17-058 

6-626 

125316 

44361864 

354 

18-814 

7-074 

85264 

24897088 

292 

17-088 

6-634 

126025 

44738875 

355 

18-841 

7-080 

85849 

25153757 

293 

17-117 

6-641 

126736 

45118016 

356 

18-867 

7-087 

86436 

25412184 

294 

17-146 

6-649 

127449 

45499293 

357 

18-894 

7-093 

87025 

25672375 

295 

17-175 

6-656 

128164 

45882712 

358 

18-920 

7-100 

87616 

25934836 

296 

17-204 

6-664 

128881 

46268279 

359 

18-947 

7-107 

88209 

26198073 

297 

17-233 

6-671 

129600 

46656000 

360 

18-973 

7-113 

88804 

26463592 

298 

17-262 

6-679 

130321 

47045831 

361 

19-000 

7-120- 

89401 

26730899 

299 

17-291 

6-686 

131044 

47437928 

362 

19-026 

7-126 

90000 

27000000 

300 

17-320 

6-694 

131769 

47832147 

363 

19-052 

7-133 

90601 

27270901 

301 

17-349 

6-701 

132496 

48228544 

364 

19-078 

7-140 

91204 

27543608 

302 

17-378 

6-709 

133225 

48627125 

365 

19-104 

7-146- 

91809 

27818127 

303 

17-406 

6-716 

133956 

49027896 

366 

19-131 

7-153 

92416 

28094464 

304 

17-435 

6-723 

134689 

49430863 

367 

19-157 

7-159- 

93025 

28372625 

305 

17-464 

6-731 

135424 

49836032 

368 

19-183 

7-166 

93636 

28652616 

306 

17-492 

6-738 

136161 

50243409 

369 

19-209 

7'172 

94249 

28934443 

307 

17-521 

6-745 

136900 

50653000 

370 

19-235 

7-17£ 

94864 

29218112 

308 

17-549 

6-753 

137641 

51064811 

371 

19-261 

7-185 

95481 

29503609 

309 

17-578 

6-760 

138384 

51478848 

372 

19-287 

7-191 

96100 

29791000 

310 

17-606 

6-767 

139129 

51895117 

373 

19-313 

7-198 

98721 

30080231 

311 

17-635 

6-775 

139876 

52313624 

374 

19-339 

7-204 

97344 

30371328 

312 

17-663 

6-782 

140625 

52734375 

375 

19-364 

7-211 

97969 

30664297 

313 

17-691 

6-789 

141376 

53157376 

376 

19-390 

7-217 

98596 

30959144 

314 

17-720 

6-796  1  142129 

53582633 

377 

19-416 

7-224 

99225 

31255875 

316 

17-748 

6-804  1  142884 

54010152 

378 

19-442 

7-230 

APPENDIX. 


699 


TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS   OF  NUMBERS— ( Continued}. 


Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

143641 

54439939 

379 

19-467 

7-236 

195364 

86350888 

442 

21-023 

7-617 

144400 

54872000 

380 

19-493 

7-243 

196249 

86938307 

443 

21-047 

7-623 

145161 

55306341 

381 

19-519 

7-249 

197136 

87528384 

444 

21-071 

7-628 

145924 

55742968 

382 

19-544 

7-255 

198025 

88121125 

445 

21-095 

7-634 

146689 

56181887 

383   19-570 

7-262 

198916 

88716536 

446 

21-118 

7-640 

147456 

56623104 

384 

19-595 

7-268 

199809 

89314623 

447 

21-142 

7-646 

148225 

57066625 

385 

19621 

7-274 

200704 

89915392 

448 

21-166 

7-651 

148996 

57512456 

386 

19-646 

7-281 

201601 

90518849 

449 

21-189 

7-657 

149769 

57960603 

387 

19-672 

7-287 

202500 

91125000 

450 

21-213 

7-663 

150544 

58411072 

388 

19-697 

7-293 

203401 

91733851 

451 

21-236 

7-668 

151321 

58863869 

389 

19-723 

7-299 

204304 

92345408 

452 

21-260 

7-674 

152100 

59319000 

390 

19-748 

7-306 

205209 

92959677 

453 

21-283 

7-680 

152881 

59776471 

391 

19-773 

7-312 

206116 

93576664 

454 

21-307 

7-685 

153664 

602S6288 

392 

19-798 

7-318 

207025 

94196375 

455 

21-330 

7-691 

154449 

60698457 

393 

19-824 

7-324 

207936 

94818816 

456 

21-354 

7-697 

155236 

61162984 

394 

19-849 

7-331 

208849 

95443993 

457 

21-377 

7-702 

156025 

61629875 

395 

19-874 

7-337 

209764 

96071912 

458 

21-400 

7-708 

156816 

62099136 

396 

19-899 

7-343 

210681 

96702579 

459 

21-424 

7-713 

157609 

62570773 

397 

19-924 

7-349 

211600 

97336000 

460 

21-447 

7-719 

158404 

63044792 

398 

19-949 

7-355 

212521 

97972181 

461 

21-470 

7-725 

159201 

63521199 

399 

19-974 

7-361 

213444 

98611128 

462 

21-494 

7-730- 

160000 

64000000 

400 

20-000 

7-368 

214369 

99252847 

463 

21-517 

7-736 

160801 

64481201 

401 

20-024 

7-374 

215296 

99897344 

464 

21-540 

7-741 

161604 

64964808 

402 

20-049 

7-380 

216225 

100544625 

465 

21-563 

7-74T 

162409 

65450827 

403 

20-074 

7-386 

217156 

101194696 

466 

21-587 

7-752 

163216 

65939264 

404 

20-099 

7-392 

218089 

101847563 

467 

21-610 

7-758 

164025 

66430125 

405 

20-124 

7-398 

219024 

102503232 

468 

21-633 

7-763 

164836 

66923416 

406. 

20-149 

7-404 

219961 

103161709 

469 

21-656 

7-769 

165649 

67419143 

407 

20-174 

7-410 

220900 

103823000 

470 

21-679 

7-774 

166464 

67917312 

408 

20-199 

7-416 

221841 

104487111 

471 

21-702 

7-780' 

167281 

68417929 

409 

20-223 

7-422 

222784 

105154048 

472 

21-725 

7-785 

168100 

68921000 

410 

20-248 

7-428 

223729 

105823817 

473 

21-748 

7-791 

168921 

69426531 

411 

20-273 

7-434 

224676 

106496424 

474 

21-771 

7-796 

169744 

69934523 

412 

20-297 

7-441 

225625 

107171875 

475 

21-794 

7-802 

170569 

70444997 

413 

20-322 

7-447 

226576 

107850176 

476 

21-817 

7-807 

171396 

70957944 

414 

20-346 

7-453 

227529 

108531333 

477 

21-840 

7-813 

172225 

71473375 

415 

20-371 

7-459 

228484 

109215352 

478 

21-863 

7-818 

173056 

71991296 

416 

20-396 

7-465 

229441 

109902239 

479 

21-886 

7-824 

173889 

72511713 

417 

20-420 

7-470 

230400 

110592000 

480 

21-908 

7-829 

174724 

73034632 

418 

20-445 

7-476 

231361 

111284641 

481 

21-931 

7-835 

175561 

73560059 

419 

20-469 

7-482 

232324 

111980168 

482 

21-954 

7-840' 

176400 

74088000 

420 

20-493 

7-488 

233289 

112678587 

483 

21-977 

7-846 

177241 

74618461 

421 

20-518 

7-494 

234256 

113379904 

484 

22-000 

7-851 

178084 

75151448 

422 

20-542 

7-500 

235225 

114084125 

485 

22-022 

7-856 

178929 

75686967 

423 

20-566 

7-506 

236196 

114791256 

486 

22-045 

7-862 

179776 

76225024 

424 

20-591 

7-512 

237169 

115501303 

487 

22-068 

7-867 

180625 

76765625 

425 

20-615 

7-518 

238144 

116214272 

488 

22-090 

7-872 

181476 

77308776 

426 

20-639 

7-524 

239121 

116930169 

489 

22-113 

7-878 

182329 

77854483 

427 

20-663 

7-530 

240100 

117649000 

490 

22-135 

7-883 

183184 

78402752 

428 

20-688 

7-536 

241081 

118370771 

491 

22-158 

7-889 

184041 

78953589 

429 

20-712 

7-541 

242064 

119095488 

492 

22-181 

7-894 

184900 

79507000 

430 

20-736 

7-547 

243049 

119S23157 

493 

22-203 

7-899 

185761 

80062991 

431 

20-760 

7-553 

244036 

120553784 

494 

22-226 

7-905 

186624 

80621568 

432 

20-784 

7-559 

245025 

121287375 

495 

22*248 

7-910 

187489 

81182737 

433 

20-808 

7-565 

246016 

122023936 

496 

22-271 

7-915 

188356 

81746504 

434 

20-832 

7-571 

247009 

122763473 

497 

22-293 

7-921 

189225 

82312875 

435 

20-856 

7-576 

248004 

123505992 

498 

22-315 

7-926 

190096 

82881856 

436 

20-880 

7-582 

249001 

124251499 

499 

22-338 

7-931 

190969 

83453453 

437 

20-904 

7-588 

250000 

125000000 

500 

22-360 

7-937 

191844 

84027672 

438 

20-928 

7-594 

251001 

125751501 

501 

22-383 

7-942 

192721 

84604519 

439 

20-952 

7-600 

252004 

126506008 

502 

22-405 

7-947 

193600 

85184000 

440 

20-976 

7-605 

253009 

127263527 

503 

22-427 

7-952 

194481 

85766121 

441 

21-000 

7-611 

254016 

128024064 

504 

22-449 

7'95& 

TOO 


APPENDIX. 


TABLE  OF  SQUAKES,  CUBES,  SQUAKE  AND  CUBE  ROOTS  OF  NUMBERS— (Continued). 


Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

255025 

128787625 

505 

22-472 

7-963 

322624 

183250432 

568 

23-832 

8-281 

256036 

129554216 

506 

22-494 

7-968 

323761 

184220009 

569 

23-853 

8-286 

257049 

130323843 

507 

22-516 

7-973 

324900 

185193000 

670 

23-874 

8-291 

258064 

131096512 

508 

22-538 

7-979 

326041 

186169411 

671 

23-895 

8-296 

259081 

131872229 

509 

22-561 

7-984 

327184 

187149248 

572 

23-916 

8-301 

260100 

132651000 

510 

22-683 

7-989 

328329 

188132517 

573 

23-937 

8-305 

261121 

133432831 

611 

22-605 

7-994 

329476 

189119224 

574 

23-958 

8-310 

262144 

134217728 

512 

22-627 

8-000 

330625 

190109375 

575 

23-979 

8-315 

263169 

135005697 

513 

22-649 

8-005 

331776 

191102976 

576 

24-000 

8-320 

264196 

135796744 

514 

22-671 

8-010 

332929 

192100033 

577 

24-020 

8-325 

265225  136590875 

515 

22-693 

8-015 

334084 

193100552 

578 

24-041 

8-329 

266256 

137388096 

516 

22-715 

8-020 

335241 

194104539 

579 

24-062 

8-334 

267289 

138188413 

517 

22-737 

8-025 

336400 

195112000 

580 

24-083 

8-339 

268324 

138991832 

618 

22-759 

8-031 

337561 

196122941 

681 

24-103 

8-344 

269361 

139798359 

519 

22-781 

8-036 

338724 

197137368 

582 

24-124 

8-349 

270400 

140608000 

520 

22-803 

8-041 

339889 

198155287 

583 

24-145 

8-353 

271441 

141420761 

521 

22-825 

8-046 

341056 

199176704 

584 

24-166 

8-358 

272484 

142236648 

522 

22-847 

8-051 

342225 

200201625 

585 

24-186 

8-363 

273529 

143055667 

623 

22-869 

8-056 

343396 

201230056 

586 

24-207 

8-368 

274576 

143877824 

524 

22-891 

8-062 

344569 

202262003 

587 

24-228 

8-372 

275625 

144703125 

525 

22-912 

8-067 

345744 

203297472 

588 

24-248 

8-377 

276676 

145531576 

526 

22-934 

8-072 

346921 

204336469 

589 

24-269 

8-382 

277729 

146363183 

527 

22-956 

8-077 

348100 

205379000 

590 

24-289 

8-387 

278784 

147197952 

528 

22-978 

8-082 

349281 

206425071 

591 

24-310 

8-391 

279841 

148035889 

529 

23-000 

8-037 

350464 

207474688 

592 

24-331 

8-396 

280900 

148877000 

530 

23-021 

8-092 

351649 

208527857 

693 

24-351 

8-401 

281961 

149721291 

531 

23-043 

8-097 

352836 

209584584 

694 

24-372 

8-406 

283024 

150568768 

532 

23-065 

8-102 

354025 

210644875 

595 

24-392 

8-410 

284089 

151419437 

633 

23-086 

8-107 

355216 

211708736 

696 

24-413 

8-415 

285156 

152273304 

534 

23-108 

8-112 

356409 

212776173 

597 

24-433 

8-420 

286225 

153130375 

535 

23-130 

8-118 

357604 

213847192 

698 

24'454 

8-424 

287296 

153990656 

636 

23-151 

8-123 

358801 

214921799 

599 

24-474 

8-429 

288369 

154854153 

637 

23-173 

8-128 

360000 

216000000 

600 

24-494 

8-434 

289444 

155720872 

538 

23-194 

8-133 

361201 

217081801 

601 

24-515 

8439 

290521 

156590819 

639 

23-216 

8-138 

362404 

218167208 

602 

24-535 

8-443 

291600 

157464000 

640 

23-237 

8-143 

363609 

219256227 

603 

24-556 

8-448 

292681 

158340421 

541 

23-259 

8-148 

364816 

220348864 

604 

24-576 

8-453 

293764 

159220088 

542 

23-280 

8-153 

366025 

221445125 

605 

24-596 

8-457 

294849 

160103007 

543 

23-302 

8-158 

367236 

222545016 

606 

24-617 

8-462 

295936 

160989184 

544 

23-323 

8-163 

368449 

223648543 

607 

24-637 

8-467 

297025 

161878625 

545 

23-345 

8-168 

369664 

224755712 

608 

24-657 

8-471 

298116 

162771336 

546 

23-366 

8-173 

370881 

225866529 

609 

24-677 

8-476 

299209 

163667323 

547 

23-388 

8-178 

372100 

226981000 

610 

24-698 

8-480 

300304 

164566592 

548 

23-409 

8-183 

373321 

228099131 

611 

24-718 

8-485 

301401 

165469149 

549 

23-430 

8-188 

374544 

229220928 

612 

24-738 

8-490 

302500 

166375000 

560 

23-452 

8-193 

375769 

230346397 

613 

24-758 

8-494 

303601 

167284151 

551 

23-473 

8-198 

376996 

231475544 

614 

24-779 

8-499 

304704 

168196608 

652 

23-494 

8-203 

378225 

232608375 

615 

24-799 

8-504 

305809 

169112377 

553 

23-515 

8-208 

379456 

233744896 

616 

24-819 

8-508 

306916 

170031464 

554 

23-537 

8-213 

380689 

234885113 

617 

24-839 

8-513 

308025 

170953875 

555 

23-558 

8-217 

381924 

236029032 

618 

24-859 

8-517 

309136 

171879616 

556 

23-579 

8-222 

383161 

237176659 

619 

24-879 

8-522 

310249 

172808693 

557 

23-600 

8-227 

384400 

238328000 

620 

24-899 

8-527 

311364 

173741112 

558 

23-622 

8-232 

385641 

239483061 

621 

24-919 

8-531 

312481 

174676879 

559 

23-643 

8-237 

386884 

240641848 

622 

24-939 

8-536 

313600 

175616000 

560 

23-664 

8-242 

388129 

241804367 

623 

24-959 

8-540 

314721 

176558481 

561 

23-685 

8-247 

389376 

242970624 

624 

24-979 

8-545 

315844 

177504328 

562 

23-706 

8-252 

390625 

244140625 

625 

25-000 

8-549 

316969 

178453547 

563 

23-727 

8-257 

391876 

245314376 

626 

25-019 

8-554 

318096 

179406144 

564 

23-748 

8-262 

393129 

246491883 

627 

25-039 

8-558 

319225 

180362125 

565 

23-769 

8-267 

394384 

247673152 

628 

25-059 

8-563 

320356 

181321496 

566 

23-790 

8-271 

395641 

248858189 

629 

25-079 

8-568 

321489 

182284263 

667 

23-811 

8-276 

396900 

250047000 

630 

25-099 

8-572 

APPENDIX. 


701 


TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS  OF  NUMBERS— (Continued}. 


Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

398161 

251239591 

631 

25-119 

8-577 

481636 

334255384 

694 

26-343 

8-853 

399424 

252435968 

632 

25-139 

8-581 

483025 

335702375 

695 

26-362 

8-857 

400689 

253636137 

633 

25-159 

8-586 

484416 

337153536   696 

26-381 

8-862 

401956 

254840104 

634 

25-179 

8-590 

485809 

338608873   697 

26-400 

8-866 

403225 

256047875 

635 

25-199 

8-595 

487204 

340068392   698 

26-419 

8-870 

404496 

257259456 

636 

25-219 

8-599 

488601 

341532099   699 

26-438 

8-874 

405769 

258474853 

637 

25-238 

8-604 

490000 

343000000 

700 

26-457 

8-879 

407044 

259694072 

638 

25-258 

8-608 

491401 

344472101 

701 

26-476 

8-883 

408321 

260917119 

639 

25-278 

8-613 

492804 

345948408 

702 

26-495 

6-887 

409600 

262144000 

640 

25-298 

8-617 

494209 

347428927   703 

26-514 

8-891 

410881 

263374721 

641 

25-317 

8-622 

495616 

348913664 

704 

26-532 

8-895 

412164 

264609288 

642 

25-337 

8-626 

497025 

350402625 

705 

26-551 

8-900 

413449 

265847707 

643 

25-357 

8-631 

498436 

351895816 

706 

26-570 

8-904 

414736 

267089984 

644 

25-377 

3-635 

499849 

353393243 

707 

26-589 

8-908 

416025 

268336125 

645 

25-396 

8-640 

501264 

354894912 

708 

26-608 

8-912 

417316 

269586136 

646 

25-416 

8-644 

502681 

356400829 

709 

26-627 

8-916 

418609 

270840023 

647 

25-436 

8-649 

504100 

357911000 

710 

26-645 

8-921 

419904 

272097792 

648 

25-455 

8-653 

505521 

359425431 

711 

26-664 

8-925 

421201 

273359449 

€49 

25-475 

8-657 

1  506944 

360944128 

712 

26-683 

8-929 

422500 

274625000 

650 

25-495 

8-662 

508369 

362467097 

713 

26-702 

8-933 

423801 

275894451 

651 

25-514 

8-666 

509796 

363994344 

714 

26-720 

8-937 

425104 

277167808 

652 

25-534 

8-671 

511225 

365525875 

715 

26-739 

8-942. 

426409 

278445077 

653 

25-553 

8-675 

512656 

367061696 

716 

26-758 

8-946 

427716 

279726264 

654 

25-573 

8-680 

514089 

368601813 

717 

26-776 

8-950 

429025 

281011375 

655 

25-592 

8'684 

515524 

370146232 

718 

26-795 

8-954 

430336 

282300416 

656 

25-612 

8-688 

516961 

371694959 

719 

26-814 

8-958 

431649 

283593393 

657 

25-632 

8-693 

518400 

373248000 

720 

26-832 

8-962 

432964 

284890312 

658 

25-651 

8-697 

519841 

374805361 

721 

26-851 

8-966 

484281 

286191179 

659 

25-670 

8*702 

521284 

676367048 

722 

26-870 

8-971 

435600 

287496000 

660 

25-690 

8-706 

522729 

377933067 

723 

26-888 

8-975 

436921 

288804781 

661 

25-709 

8-710 

524176 

379503424 

724 

26-907 

8-979 

438244 

290117528 

662 

25-729 

8-715 

525625 

381078125 

725 

26-925 

8-983 

439569 

291434247 

663 

25*748 

8*719 

527076 

382657176 

726 

26-944 

8-987 

440896 

292754944 

664 

25-768 

8-724 

528529 

384240583 

727 

26-962 

8991 

442225 

294079625 

665 

25-787 

8-728 

529984 

385828352 

728 

26-981 

8-995 

443556 

295408296 

666 

25-806 

8'732 

531441 

387420489 

729 

27-000 

9-000 

444889 

296740963 

667 

25-826 

8*737 

532900 

389017000 

730 

27*018 

9-004 

446224 

298077632 

668 

25-845 

8-741 

534361 

390617891 

731 

27-037 

9-008 

447561 

299418309 

669 

25-865 

8-745 

535824 

392223168 

732 

27-055 

9-012 

448900 

300763000 

670 

25-884 

8-750 

5372b9 

393832837 

733 

27-073 

9-016 

450241 

302111711 

671 

25-903 

8-754 

538756 

395446904 

734 

27-092 

9-020 

451584 

303464448 

672 

25-922 

8-759 

540225 

397065375 

735 

27-110 

9/024 

452929 

304821217 

673 

25-942 

8-763 

541696 

398688256 

736 

27-129 

9-028 

454276 

306182024 

674 

25-961 

8'767 

543169 

400315553 

787 

27*147 

9-032 

455625 

307546875 

675 

25-980 

8-772 

544644 

401947272 

738 

27-166 

9-036 

456976 

308915776 

676 

26-000 

8-776 

546121 

403583419 

739 

27-184 

9-040 

458329 

310288733 

677 

26-019 

8-780 

547600 

405224000 

740 

27-202 

9-045 

459684 

311665752 

678 

26-038 

8-785 

549081 

406869021 

741 

27-221 

9-049 

461041 

313046839 

679 

26-057 

8-789 

550564 

408518488 

742 

27-239 

9-053 

462400 

314432000 

680 

26-076 

8-793 

552049 

410172407 

743 

27-258 

9-057 

463761 

315821241 

681 

26-095 

8-797 

553536 

411830784 

744 

27-276 

9-061 

465124 

317214568 

682 

26-115 

8-802 

555025 

413493625 

745 

27-294 

9-065 

466489 

318611987 

683 

26-134 

8-806 

556516 

415160936 

746 

27-313 

9-069 

467856 

320013504 

684 

26-153 

8-810 

558009 

416832723 

747 

27-331 

9-073 

469225 

321419125 

685 

26-172 

8-815 

559504 

418508992 

748 

27-349 

9-077 

470596 

322828856 

686 

26-191 

8-819 

561001 

420189749 

749 

27-367 

9-081 

471969 

324242703 

687 

26-210 

8-823 

562500 

421875000 

750 

27-386 

9-085 

473344 

325660672 

688 

26-229 

8-828 

564001 

423564751 

751 

27-404 

9-089 

474721 

327082769 

689 

26-248 

8-832 

565504 

425259008 

752 

27-422 

9-093 

476100 

328o09000 

690 

26-267 

8-836 

567009 

426957777 

753 

27-440 

9-097 

477481 

329939371 

691 

26*286 

8-840 

568516 

428661064 

754 

27-459 

9-101 

478864 

331373888 

692 

26-305 

8-845 

570025 

430368875 

755 

27-477 

9-105 

480249 

332812557 

693 

26-324 

8-849 

571536 

432081216 

756 

27-495 

9-109 

702 


APPENDIX. 


TABLE  OF  SQUARES,  CUBES,  SQUAEE  AND  CUBE  EOOTS  OF  NUMBERS— (Continued). 


Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

673049 

433798093 

757 

27-513 

9-113 

672400 

551368000 

820 

28-635 

9-359 

574564 

435519512 

758 

27-531 

9-117 

674041 

553387661 

821 

28-653 

9-363 

576081 

437245479 

759 

27-549 

9-121 

675684 

555412248 

822 

28-670 

9-367 

577600 

438976000 

760 

27-568 

9-125 

677329 

557441767 

823 

28-687 

9-371 

579121 

440711081 

761 

27-586 

9-129 

678976 

559476224 

824 

28-705 

9-375 

580644 

442450728 

762 

27-604 

9-133 

680625 

561515625 

825 

28-722 

9-378 

582169 

444194947 

763 

27-622 

9-137 

682276 

563559976 

826 

28-740 

9-382 

583696 

445943744 

764 

27-640 

9-141 

683929 

565609283 

827 

28-757 

9-386 

585225 

447697125 

765 

27-658 

9-145 

685584 

567663552 

828 

28-774 

9-390 

586756 

449455096 

766 

27-676 

9-149 

687241 

569722789 

829 

28-792 

9-394 

588289 

451217663 

767 

27-694 

9-153 

688900 

571787000 

830 

28-809 

9-397 

589824  . 

452984832 

768 

27-712 

9-157 

690561 

573856191 

831 

28-827 

9-401 

591361 

454756609 

769 

27-730 

9-161 

692224 

575930368 

832 

28-844 

9-405 

692900 

456533000 

770 

27-748 

9-165 

693889 

578009537 

833 

28-861 

9-409 

594441 

458314011 

771 

27-766 

9-169 

695556 

580093704 

834 

28-879 

9-412 

595984 

460099648 

772 

27-784 

9-173 

697225 

582182875 

835 

28-896 

9-416 

697529 

461889917 

773 

27-802 

9-177 

698896 

584277056 

836 

28-913 

9-420 

699076 

463684824 

774 

27-820 

9-181 

700569 

586376253 

837 

28-930 

9-424 

600625 

465484375 

775 

27-833 

9-185 

702244 

588480472 

838 

28-948 

9-427 

602176 

467288576 

776 

27-856 

9-189 

703921 

590589719 

839 

28-965 

9-431 

603729 

469097433 

777 

27-874 

9-193 

705600 

592704000 

840 

28-982 

9-435 

605284 

470910952 

778 

27-892 

9-197 

707281 

594823321 

841 

29-000 

9-439 

606841 

472729139 

779 

27-910 

9-201 

708964 

596947688 

842 

29-017 

9-442 

608400 

474552000 

780 

27-928 

9-205 

710649 

599077107 

843 

29-034 

9-446 

609961 

476379541 

781 

27-946 

9-209 

712336 

601211584 

844 

29-051 

9-450 

611524 

478211768 

782 

27-964 

9-213 

714025 

603351125 

845 

29-068 

9-454 

613089 

480048687 

783 

27-982 

9-216 

715716 

605495736 

846 

29-086 

9-457 

614656 

481890304 

784 

28-000 

9-220 

717409 

607645423 

847 

29-103 

9-461 

616225 

483736625 

785 

28-017 

9-224 

719104 

609800192 

848 

29-120 

9-465 

617796 

485587656 

786 

28-035 

9-228- 

720801 

611960049 

849 

29-137 

9-468 

619369 

487443403 

787 

28-053 

9-232 

722500 

614125000 

850 

29-154 

9-472 

620944 

489303872 

788 

28-071 

9-236 

724201 

616295051 

851 

29-171 

9-476 

622521 

491169069 

789 

28-089 

9-240 

725904 

618470208 

852 

29-189 

9-480 

624100 

493039000 

790 

28-106 

9-244 

727609 

620650477 

853 

29-206 

9-483 

625681 

494913671 

791 

28-124 

9-248 

729316 

622835864 

854 

29-223 

9-487 

627264 

496793088 

792 

28-142 

9-252 

731025 

625026375 

855 

29-240 

9-491 

628849 

498677257 

793 

28-160 

9-256 

732736 

627222016 

856 

29-257 

9494 

630436 

500566184 

794 

28-178 

9-259 

734449 

629422793 

857 

29-274 

9-498 

632025 

502459875 

795 

28-195 

9-263 

736164 

631628712 

858 

29-291 

9-502 

633616 

504358336 

796 

28-213 

9"267 

737881 

633839779 

859 

.  29-308 

9-505 

635209 

506261573 

797 

28231 

9-271 

739600 

636056000 

860 

29-325 

9-509 

636804 

508169592 

798 

28-248 

9-275 

741321 

638277381 

861 

29-342 

9-513 

638401 

510082399 

799 

28-266 

9-279 

743044 

640503928 

862 

29-359 

9-517 

640000 

512000000 

800 

28-284 

9-283 

744769 

642735647 

863 

29-376 

9-520 

•641601 

513922401 

801 

28-301 

9-287 

746496 

644972544 

864 

29-393 

9-524 

643204 

515849608 

802 

28-319 

9-290 

748225 

647214625 

865 

29-410 

9-528 

644809 

517781627 

803 

28-337 

9-294 

749956 

649461896 

866 

29-427 

9-531 

646416 

519718464 

804 

28-354 

9-298 

751689 

651714363 

867 

29-444 

9-535 

648025 

521660125 

805 

28-372 

9-302 

753424 

653972032 

868 

29-461 

9-539 

649636 

523606616 

806 

28-390 

9-306 

755161 

656234909 

869 

29-478 

9  542 

651249 

525557943 

807 

28-407 

9-310 

756900 

658503000 

870 

29-495 

9-546 

652864 

527514112 

808 

28-425 

9-314 

758641 

660776311 

871 

29-512 

9-650 

654481 

529475129 

809 

28-442 

9-317 

760384 

663054848 

872 

29-529 

9-553 

656100 

531441000 

810 

28-460 

9-321 

762129 

665338617 

873 

29516 

9-557 

657721 

533411731 

811 

28-478 

9-325 

763876 

667627624 

874 

29-r><>::! 

9-561 

659344 

535387328 

812 

28-495 

9-329 

765625 

669921875 

875 

29-580 

9-564 

660969 

537367797 

813 

28-513 

9-333 

767376 

672221376 

876 

29-597 

9'568 

662596 

539353144 

814 

28-530 

9-337 

769129 

674526133 

877 

29-614 

9-571 

664225 

541343375 

815 

28-548 

9-340 

770884 

676836152 

878 

29-631 

9-575 

665856 

543338496 

816 

28-565 

9-344 

772641 

679151439 

879 

29-647 

9-579 

667489 

545338513 

817 

28-583 

9-348 

774400 

681472000 

880 

29-664 

9-582 

669124 

547343432 

818 

28-600 

9-352 

776161 

683797841 

881 

29-681 

9-586 

£70761 

549353259 

819 

28-618 

9-356  ! 

777924 

686128968 

882 

29-698 

9-590 

APPENDIX. 


703 


TABLE  OF  SQUARES,  CUBES,  SQUARE  AND  CUBE  BOOTS  OF  NUMBERS— ( Continued). 


Squares. 

Cubes. 

No. 

Square 
roots. 

Cube 
roots. 

Squares. 

Cubes. 

Square 

Cube 

roots. 

roots. 

779689 

688465387 

883 

29-715 

9-593 

894916 

846590586 

946 

30-757 

9-816 

781456 

690807104 

884 

29-732 

9-597 

896808 

849278m 

947 

30-773 

9-820 

783225 

693154125 

885 

29-748 

9-600 

898704 

851971392 

948 

30-789 

9-823 

784996 

695506456 

886 

29-765 

9-604 

900601 

854670349 

949 

30-805 

9-827 

786769 

697864103 

887 

29-782 

9-608 

902500 

857375000 

950 

30-822 

9-830 

788544 

700227072 

888 

29-799 

9-611 

904401 

860085351 

951 

30-838 

9-833 

790321 

702595369 

889 

29-816 

9-615 

906304 

862801408 

952 

30-854 

9-837 

792100 

704969000 

890 

29-832 

9-619 

908209 

865523177 

953 

30-870 

9-840 

793881 

707347971 

891 

29-849 

9-622 

910116 

868250664 

954 

30-886 

9-844 

795664 

709732288 

892 

29-866 

9-626 

912025 

870983875 

955 

30-903 

9-847 

797449 

712121957 

893 

29-883 

9-629 

913936 

873722816 

956 

30-919 

9-851 

799236 

714516984 

894 

29-899 

9-633 

915849 

876467493 

957 

30-935 

9-854 

801025 

716917375 

895 

29-916 

9-636 

917764 

879217912 

958 

30-951 

9-857 

802816 

719323136 

896 

29-933 

9-640 

919681 

881974079 

959 

30-967 

9-861 

804609 

721734273 

897 

29-949 

9-644 

921600 

884736000 

960 

30-983 

9-864 

806404 

724150792 

898 

29-966 

9-647 

923521 

887503681 

961 

31-000 

9-868 

808201 

726572699 

899 

29-983 

9-651 

925444 

890277128 

962 

31-016 

9-871 

810000 

729000000 

900 

30-000 

9-654 

927369 

893056347 

963 

31-032 

9-875 

811801 

731432701 

901 

30-016 

9-658 

929296 

895841344 

964 

31-048 

9-878 

813604 

733870808 

902 

30-033 

9-662 

931225 

898632125 

965 

31-064 

9-881 

815409 

736314327 

903 

30-049 

9-665 

933156 

901428696 

966 

31-080 

9-885 

817216 

738763264 

904 

30-066 

9-669 

935089 

904231063 

967 

31-096 

9-888 

819025 

741217625 

905 

30-083 

9-672 

937024 

907039232 

968 

31-112 

9-892 

820836 

743677416 

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30-099 

9-676 

938961 

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969 

31-128 

9-895 

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30-116 

9-679 

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31-144 

9-898 

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30-133 

9-683 

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31-160 

9-902 

826281 

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30-149 

9-686 

944784 

918330048 

972 

31-176 

9-905 

828100 

753571000 

910 

30-166 

9-690 

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31-192 

9-909 

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30-182 

9-694 

948676 

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31-208 

9-912 

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30-199 

9-697 

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926859375 

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31-224 

9-915 

833569 

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30-215 

9-701 

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31-240 

9-919 

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30-232 

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31-256 

9-922 

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9-708 

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31-272 

9-926 

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30-265 

9-711 

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31-288 

9-929 

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30-282 

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31-320 

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9-722 

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31-336 

9-939 

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LATITUDES   AND   DEPARTURES. 


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APPENDIX. 


705 


LATITUDES    AND    DEPARTURES. 


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1-556 

3-420 

2-075 

4-275 

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0-853 

0-522 

1-705 

1-045 

2-558 

1-567 

3-411 

2-090 

4-263 

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31*  ! 

0-850 

0-526 

1-701 

1-052 

2-551 

1-579 

3-401 

2-105 

4-252 

58i 

32° 

0-848 

0-530 

1-696 

1-060 

2-544 

1-590 

3-392 

2-120 

4-240 

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32i 

0-846 

0-534 

1-691 

1-067 

2-537 

1-601 

3-383 

2-134 

4-229 

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32* 

0-843 

0-537 

1-687 

1-075 

2-530 

1-612 

3-374 

2-149 

4-217 

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0-841 

0-541 

1-682 

1-082 

2-523 

1-623 

3-364 

2-164 

4-205 

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33° 

0-839 

0-545 

1-677 

1-089 

2-516 

1-634 

3-355 

2-179 

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0-836 

0-548 

1-673 

1-097 

2-509 

1-645 

3-345 

2-193 

4-181 

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0-834 

0-552 

1-668 

1-104 

2-502 

1-656 

3-336 

2-208 

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0-831 

0-556 

1-663 

1-111 

2-494 

1-667 

3-326 

2-222 

4-157 

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0-829 

0-559 

1-658 

1-118 

2-487 

1-678 

3-316 

2-237 

4-145 

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0-827 

0-563 

1-653 

1-126 

2-480 

1-688 

3-306 

2-251 

4-133 

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0-824 

0-566 

1-648 

1-133 

2-472 

1-699 

3-297 

2-266 

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0-570 

1-643 

1-140 

2-465 

1-710 

3-287 

2-280 

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0-574 

1-638 

1-147 

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3-277 

2-294 

4-096 

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0-817 

0-577 

1-633 

1-154 

2-450 

1-731 

3-267 

2-309 

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354 

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0-581 

1-628 

1-161 

2-442 

1-742 

3-257 

2-323 

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0-812 

0-584 

1-623 

1-168 

2-435 

1-753 

3-246 

2-337 

4-058 

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0-809 

0-588 

1-618 

1-176 

2-427 

1-763 

3-236 

2-351 

4-045 

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0-806 

0-591 

1-613 

1-183 

2-419 

1-774 

3-226 

2-365 

4-032 

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0-804 

0-595 

1-608 

1-190 

2-412 

1-784 

3-215 

2-379 

4-019 

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0-801 

0-598 

1-603 

1-197 

2-404 

1-795 

3-205 

2-393 

4-006 

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0-799 

0-602 

1-597 

1-204 

2-396 

1-805 

3-195 

2-407 

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0-796 

0-605 

1-592 

1-211 

2-388 

1-816 

3-184 

2-421 

3-980 

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0-793 

0-609 

1-587 

1-218 

2-380 

1-826 

3-173 

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0-791 

0-612 

1-531 

1-224 

2-372 

1-837 

3-163 

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0-788 

0-616 

1-576 

1-231 

2-364 

1-847 

3-152 

2-463 

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38± 

0-785 

0-619 

1-571 

1-238 

2-356 

1-857 

3-141 

2-476 

3-927 

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0-783 

0-623 

1-565 

1-245 

2-348 

1-868 

3-130 

2-490 

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0-780 

0-626 

1-560 

1-252 

2-340 

1-878 

3-120 

2-604 

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1-259 

2-331 

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3-109 

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0-633 

1-549 

1-255 

2-323 

1-898 

3-098 

2-i31 

3-872 

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1-543 

1-272 

2-315 

1-908 

3-086 

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0-639 

1-538 

1-279 

2-307 

1-918 

3-075 

2-558 

3-844 

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0-766 

0-643 

1-532 

1-286 

2-298 

1-928 

3-064 

2-571 

3-830 

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0-763 

0-646 

1-526 

1-292 

2-290 

1-938 

3-053 

2-584 

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0-760 

0-649 

1-521 

1-299 

2-281 

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0-653 

1-515 

1-306 

2-273 

1-958 

3-030 

2-611 

3-788 

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0-656 

1-509 

1-312 

2-264 

1-968 

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2-624 

3-774 

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0-752 

0-659 

1-504 

1-319 

2-256 

1978 

3-007 

2-637 

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0-749 

0-663 

1-498 

1-325 

2-247 

1-988 

2-996 

2-650 

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0-746 

0-666 

1-492 

1-332 

2-238 

1-998 

2-984 

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0-743 

0-669 

1-486 

1-338 

2-229 

2-007 

2-973 

2-677 

3-716 

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0-740 

0-672 

1-480 

1-345 

2-221 

2-017 

2-961 

2-689 

3-701 

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0-737 

0-676 

1-475 

1-351 

2-212 

2-027 

2-949 

2-702 

3-686 

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42* 

0-734 

0-679 

1-469 

1-358 

2-203 

2-036 

2-937 

2-715 

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0-731 

0-682 

1-463 

1-364 

2-194 

2-046 

2-925 

2-728 

3-657 

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0-728 

0-685 

1-457 

1-370 

2-185 

2-056 

2-913 

2-741 

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0-725 

0-688 

1-451 

1-377 

2-176 

2-065 

2-901 

2-753 

3-627 

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0-722 

0-692 

1-445 

1-383 

2-167 

2-075 

2-889 

2-766 

3-612 

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0-719 

0-695 

1-439 

1-389 

2-158 

2-084 

2-877 

2-779 

3-597 

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0-716 

0-698 

1-433 

1-396 

2-149 

2-093 

2-S65 

2-791 

3-582 

46* 

44* 

0-713 

0-701 

1-427 

1-402 

2-140 

2-103    i 

2-853 

2-804 

3-566 

45i 

44f 

0-710 

0-704 

1-420 

1-408 

2-131 

2-112 

2-841 

2-816    ! 

3-551 

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45f 

0-707 

1-707 

1-414 

1-414 

2-121 

2-121 

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2-828 

3-536 

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Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

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Dep. 

Lat. 

Dep. 

.5 

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1 

9 

3 

4 

ft 

M 

APPENDIX. 


709 


LATITUDES    AND   DEPARTURES. 


sp 

5 

6 

7 

9 

9 

£ 

I 

Dep. 

Lat.          Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

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I 

30° 

2-500 

5-196 

3-000 

6-062 

3-500 

i    6-928 

4-000 

7-794 

4-500 

60° 

301 

2-519 

5-183 

3-023 

6-047 

3-526 

:    6-911 

4-030 

7-775 

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2-538 

5'170 

3-045 

I     6-031 

3-553 

!    6-893 

4-060 

7-755 

4-568 

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2-556 

5-156 

3-068 

6016 

3-579 

6-875 

4-090 

7735 

4-602 

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2-575 

5-143 

3-090 

6-000 

3-605 

6-857 

4-120 

7-715 

4-635 

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sil 

2-594 

5-129 

3  113 

5-984 

3-631 

6-839 

4-150 

7-694 

4-669 

58f 

811 

2-612 

5-116 

3-135 

5-968 

3-657 

6-821 

4-180 

7-674 

4-702 

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31f 

2-631 

5-102 

3-157 

5-952 

3-683 

6-803 

4-210 

7-653 

4-736 

581 

32° 

2-650 

5-088 

3-180 

5-936 

3-709 

6-784 

4-239 

7-632 

4-769 

58° 

321 

2-668 

5-074 

3-202 

5-920 

3-735 

6-766 

4-269 

7-612 

4-802 

57f 

«2J 

2-686 

5-060 

3-224 

5-904 

3-761 

6-747 

4-298 

7-591 

4-836 

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32f 

2-705 

5-046 

3-246 

5-887 

3-787 

6728 

4-328 

7-569 

4-869 

571 

33° 

2-723 

5-032 

3"268 

5-871 

3-812 

6-709 

4-357 

7-548 

4-902 

5T° 

331 

2-741 

5-018 

3-290 

5-854 

8-838 

6-690 

4-386 

7-527 

4-935 

56£ 

33! 

2-760 

5-003 

3-312 

5-837 

3-864 

6-671 

4-416 

7-505 

4-967 

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33| 

2-778 

4-989 

3333 

5-820 

3-889 

6-652 

4-445 

7-483 

5-000 

56i 

34° 

2-796 

4-974 

3355 

6-803 

3-914 

6-632     '    4-474 

7-461 

5-033 

56° 

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2-814 

4-960 

3-377 

5-786 

3-940 

i     6-613        4-502 

7-439        5-065 

55£ 

34! 

2-832 

4-945 

3-398 

5-769 

3-965 

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7'417        509*8 

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34£ 

2-850 

4-930 

3-420 

5-752 

8-990 

i     6-573        4-560 

7-396        5-130 

551 

35° 

2-868 

4-915 

3-441 

5-734 

4-015 

6-553 

4-589 

7-372        5-162 

55° 

351 

2-886 

4-900 

3-463 

5-716 

4-040 

6-533 

4-617 

7-350        5-194 

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35! 

2-904 

4-885 

3-484 

5-699 

4-065 

6-513 

4-646 

7-327 

5-226        54! 

35J 

2-921 

4-869 

3-505 

5-681 

4-090 

6-493 

4-674 

7-304 

5-268 

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36° 

2-939 

4-854 

3-527 

5-663 

4-115 

6-472 

4-702 

7-281 

5-290 

54f 

361 

2-957 

4-839 

3-548 

5-645 

4-139 

6-452 

4-730 

7'2f8 

5-322 

53f 

36^ 

2974 

4-823 

3-569 

5-627 

4-164 

6-431 

4-759 

7-235 

6-353 

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36f 

2-992 

4-808 

3-590 

5-6<>9 

4-188 

6-410 

4-787 

7-211 

5-385 

£3! 

3T° 

3-009 

4-792 

3-611 

5-590 

4-213 

6-389 

4-815 

7-188 

5-416 

53° 

371 

3-026 

4-776 

3-632 

5-572 

4-237 

6-368 

4-842 

7-164 

5-448 

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37! 

3-044 

4-760 

3-653 

5-554 

4-261 

6-347 

4-870 

7-140 

5-479 

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87| 

3-061 

4-744 

3-673 

5-535 

4-286 

6-326 

4-898 

7-116 

5-510 

621 

38° 

3-078 

4-728 

3-694 

5-516 

4-310 

6-304 

4-925 

7-092 

5-541 

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381 

3-095 

4-712 

3-715 

5-497 

4-334 

6-283 

4-953 

7-068 

5-572 

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38£' 

3-113 

4-696 

3-735 

5-478 

4-358 

6-261 

4-980 

7-043 

5-603 

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3-130 

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3-756 

5-459 

4-381 

6-239 

5-007 

7-019 

5-633 

511 

393 

3-147 

4-663 

3-776 

5-440 

4-405 

6-217 

6-035 

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5-664 

51° 

391 

3-164 

4-646 

3-796 

5-421 

4-429 

6-195 

6-062 

6-970 

6-694 

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39! 

3-180 

4-630 

3-816 

5-401 

4-453 

6-173 

6-089 

6-945 

5-725 

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3-197 

4-613 

3-837 

5-382 

4-476 

6-151 

5-116 

6-920 

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40°        3-214 

4-596 

3-857 

6-362 

4-500 

6-128 

5-142 

6-894 

5-785 

50° 

401 

3-231 

4-579 

3-877 

5-343 

4-523 

6-106 

5-169 

6-869 

5-816      i  49£ 

40! 

3-247 

4-562 

3-897 

6-323 

4-546 

6-083 

5-196 

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3-917 

5-303 

4-569 

6-061 

6-222 

6-818 

5-875 

491 

41° 

3-280 

4-528 

3-936 

6-283 

4-592 

6-038 

5-248 

6-792 

5-905 

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411 

3-297 

4-511 

3-956 

6-263 

4-615 

6-015 

5-276 

6-767 

5-934 

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41! 

3313 

4-494 

3-976 

6-243 

4-638 

6-992 

6-301 

6-741 

5-964 

48! 

41* 

3-329 

4-476 

3-995 

5-222 

4-661 

5-968 

6-327 

6-715 

5-993 

481 

42° 

3-346 

4-459 

4-015 

5-202 

4-684 

5-945 

5-363 

6-688 

6-022 

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421 

3-362 

4-441 

4-034 

6-182 

4-707 

5-922 

5-379 

6-662 

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3-378 

4-424 

4-054 

6-161 

4-729 

5-898 

5-405 

6-635 

6'080 

47! 

42f 

3-394 

4-406 

4-073 

6-140 

4-752 

5-875 

6-430 

6-609 

6-109    ! 

471 

43° 

3-410 

4-388 

4-092 

5-119 

4-774 

5-851 

6-456    i 

6-582 

6-138    i 

47° 

431 

3-426 

4-370 

4-111 

5-099 

4-796 

5-827 

5-481    ! 

6-555 

6167 

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43! 

3-442 

4-352 

4-130 

6-078 

4-818 

5-803 

5-507 

6-628 

6-195 

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43| 

3-458 

4-334 

4-149 

5-057 

4-841 

5-779 

6-532 

6-501 

6-224 

461 

44° 

3-473 

4-316 

4-168 

5-035 

4-863 

5-755 

5-557 

6-474 

6-252 

46° 

441 

3-489 

4-298 

4-187 

5-014 

4-885 

5-730 

5-582 

6-447 

6-280 

45f 

44! 

3-505 

4-280 

4-206 

4-993 

4-906 

5-706 

5-607 

6-419 

6-308 

45! 

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3-520 

4-261 

4-224 

4-971 

4-928 

5-681 

6-632 

6-392 

6-336    1 

451 

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3-536 

4-243 

4-243 

4.950 

4-950 

6-657 

5-657 

6-364 

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Lat. 

Dep. 

Lat. 

Dep.         Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

£ 

1 

5 

6 

7 

8 

9 

i 

710 


APPENDIX. 


NATURAL,     SINES     AND    COSINES. 


0° 

1° 

rj^Q 

3° 

40 

/ 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine.. 

Cosine. 

Sine. 

Cosine. 

r 

0 

00000 

Unit. 

01745 

99985 

03490 

99939 

05234 

99863 

06976 

99756 

60 

1 

00029 

Unit. 

01774 

99984 

03519 

99938 

05263 

99861 

07005 

99754 

59 

2 

00058 

Unit. 

01803 

99984 

03548 

99937  i 

05292 

99860 

07034 

99752 

58 

3 

00087 

Unit. 

01832 

99983 

03577 

99936 

05321 

99858 

07063 

99750 

57 

4 

00116 

Unit. 

01862 

99983 

03606 

99935 

05350 

99857 

07092 

99748 

56 

5 

00145 

Unit. 

01891 

99982 

03635 

99934 

05379 

99855 

07121 

99746 

55 

6 

00175 

Unit. 

01920 

99982 

03664 

99933 

05408 

99854  > 

07150 

99744 

54 

7 

00204 

Unit. 

01949 

99981 

03693 

99932 

05437 

99852 

07179 

99742 

53 

8 

00233 

Unit. 

01978 

99980  ! 

03723 

99931 

05466 

99851  | 

07208 

99740 

52 

9 

00262 

Unit. 

02007 

99980 

03752 

99930 

05495 

99849 

07237 

99738 

51 

10 

00291 

Unit. 

02036 

99979 

03781 

99929 

05524 

99847 

07266 

99736 

50 

11 

00320 

99999 

02065 

99979 

03810 

99927 

05553 

99846 

07295 

99734 

49 

12 

00349 

99999 

02094 

99978 

03839 

99926 

05582 

99844 

07324 

99731 

48 

13 

00378 

99999 

02123 

99977 

03868 

99925 

05611 

99842 

07353 

99729 

47 

14 

00407 

99999 

02152 

99977 

03897 

99924 

05640 

99841 

07382 

99727 

46 

15 

00436 

99999 

02181 

99976 

03926 

99923 

05669 

99839 

07411 

99725 

45 

16 

00465 

99999 

02211 

99976 

03955 

99922 

05698 

99838 

07440 

99723 

44 

17 

00495 

99999 

02240 

99975 

03984 

99921 

05727 

99836 

07469 

99721 

43 

18 

00524 

99999 

02269 

99974 

04013 

99919 

05756 

99834 

07498 

99719 

42, 

19 

00553 

99998 

02298 

99974 

04042 

99918 

05785 

99833 

07527 

99716 

41 

20   00582 

99998 

02327 

99973 

04071 

99917 

05814 

99831 

07556 

99714 

40 

21   00611 

99998 

02356 

99972 

04100 

99916 

05844 

99829 

07585 

99712 

39 

22 

00640 

99998 

02385 

99972 

04129 

99915 

05873 

99827 

07614 

99710 

38 

23 

00669 

99998 

02414 

99971  i 

04159 

99913 

05902 

99826 

07643 

99708 

37 

24 

00698 

99998 

02443 

99970  i 

04188 

99912 

05931 

99824 

07672 

99705 

3d 

25 

00727 

99997 

02472 

99969 

04217 

99911 

05960 

99822 

07701 

99703 

35 

26 

00756 

99997 

02501 

99969 

04246 

99910 

05989 

99821 

07730 

99701 

34 

27 

00785 

99997 

02530 

99968 

04275 

99909 

06018 

99819 

07759 

99699 

33 

28 

00814 

99997 

02560 

99967 

04304 

99907 

06047 

99817 

07788 

99696 

32 

29 

00844 

99996 

02589 

99966 

04333 

99906 

06076 

99815 

07817 

99694 

31 

30 

00873 

99996 

02618 

99966 

04362 

99905 

06105 

99813 

07846 

99692 

30- 

31 

00902 

99996 

02647 

99065 

04391 

99904 

06134 

99812 

07875 

99689 

29 

32 

00931 

99996 

02676 

99964 

04420 

99902 

06163 

99810 

07904 

99687 

2& 

33 

00960 

99995 

02705 

99963 

044-19 

99901 

06192 

99808 

07933 

99685 

27 

34 

00989 

99995 

02734 

99963 

04478 

99900 

06221 

99806 

07962 

99683 

26 

35 

01018 

99995 

02763 

99962 

04507 

99898 

06250 

99804 

07991 

99680 

25 

36 

01047 

99995 

02792 

99961 

04536 

99897 

06279 

99803 

08020 

99678 

24 

37 

01076 

99994 

02821 

99960  ; 

04565 

99896 

06308 

99801 

08049 

99676 

23 

38 

01105 

99994 

02850 

99959 

04594 

99894 

06337 

99799 

08078 

99673 

22. 

39 

01134 

99994 

02879 

99959 

04623 

99893 

06366 

99797 

08107 

99671 

21 

40 

01164 

99993 

02908 

99958 

04653 

99892 

06395 

99795 

08136 

99668 

20 

41 

01193 

99993 

•  02938 

99957 

04682 

99890 

06424 

99793 

08165 

99666 

19 

42 

01222 

99993 

02967 

99956 

04711 

99889 

06453 

99792 

08194 

99664 

18 

43 

01251 

99992 

02996 

99955  : 

.  04740 

99888 

06482 

99790 

08223 

99661 

17 

44 

01280 

99992 

03025 

99954 

04769 

99886 

06511 

99788  i 

08252 

99659 

16 

45 

01309 

99991 

03054 

99953 

04798 

99885 

06540 

99786 

08281 

99657 

15 

46 

01338 

99991 

03083 

99952 

04827 

99883 

06569 

99784 

08310 

99654 

14 

47 

01367 

99991 

03112 

99952 

04856 

99882 

06598 

99782 

08339 

99652 

13 

48 

01396 

99990 

03141 

99951 

04885 

99881 

06627 

99780 

08368 

99649 

12: 

49 

01425 

99990 

03170 

99950 

04914 

99879 

06656 

99778 

08397 

99647 

11 

50 

01454 

99989 

03199 

99949 

04943 

99878 

06685 

99776 

08426 

99644 

10 

51 

01483 

99989 

03228 

99948 

04972 

99876 

06714 

99774 

08455 

99642 

9 

52 

01513 

99989 

03257 

99947 

05001 

99875 

06743 

99772 

08484 

99639 

8 

53 

01542 

99988   03286 

99946 

05030 

99873 

06773 

99770 

08513 

99637 

7 

54 

01571 

99988   03316 

99945 

05059 

99872 

06802 

99768 

08542 

99635 

6 

55 

01600 

99987  i  03345 

99944  I  05088 

99870 

06831 

99766 

08571 

99632 

5 

56 

01629 

99987 

03374 

99943 

05117 

99869 

06860 

99764 

08600 

99630 

4 

57 

01658 

99986 

03403 

99942 

05146 

99867 

06889 

99762 

08629 

99627 

3 

58 

01687 

99986 

03432 

99941 

05175 

99866 

06918 

99760 

08658 

99625 

2 

59 

01716 

99985 

03461 

99940 

05205 

99864 

06947 

99758 

08687 

99622 

1 

60 

01745 

99985 

03490 

99939 

05234 

99863 

06976 

99756 

08716 

99619 

0 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

/ 

80° 

88° 

87° 

86° 

85° 

/ 

APPENDIX. 


711 


NATURAL.     SIXES     AND    COSINES. 


£ 

> 

C 

J° 

9 

ro 

£ 

*° 

« 

)° 

/ 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

/ 

0 

08716 

99619 

10453 

99452 

12187 

99255 

13917 

99027 

15643 

98769 

60 

1 

08745 

99617 

10482 

99449 

12216 

99251 

13946 

99023 

15672 

98764 

59 

2 

08774 

99614 

10511 

99446 

12245 

99248 

13975 

99019 

15701 

98760 

58 

3 

08803 

99612 

10540 

99443 

12274 

99244 

14004 

99015 

15730 

98755 

57 

4 

08831 

99609 

10569 

99440 

12302 

99240 

14033 

99011 

15758 

98751 

56 

5 

08860 

99607 

10597 

99437 

12331 

99237 

14061 

99006 

15787 

98746 

55 

6 

08889 

99604 

10626 

99434 

12360 

99233 

14090 

99002 

15816 

98741 

54* 

7 

08918 

99602 

10655 

99431 

12389 

99230 

14119 

98998 

15845 

98737 

53 

8 

08947 

99599 

10684 

99428 

12418 

99226 

14148 

98994 

15873 

98732 

52 

9 

08976 

99596 

10713 

99424 

12447 

99222 

14177 

98990 

15902 

98728 

51 

10 

09005 

99594 

10742 

99421 

12476 

99219 

14205 

98986 

15931 

98723 

50 

11 

09034 

99591 

10771 

99418 

12504 

99215 

14234 

98982 

15959 

98718 

49 

12 

09063 

99588 

10800 

99415 

12533 

99211 

14263 

98978 

15988 

98714 

48 

13 

09092 

99586 

10829 

99412 

12562 

99208 

14292 

98973 

16017 

98709 

47 

14 

09121 

99583 

10858 

99409 

12591 

99204 

14320 

98969 

16046 

98704 

46 

15 

09150 

99580 

10887 

99406 

12620 

99200 

14349 

98965 

16074 

98700 

45 

16 

09179 

99578 

10916 

99402 

12649 

99197 

14378 

98961 

16103 

98695 

44 

17 

09208 

99575 

10945 

99399 

12678 

99193 

14407 

98957 

16132 

98690 

43 

18 

09237 

99572 

10973 

99396 

12706 

99189 

14436 

98953 

16160 

98686 

42 

19 

09266 

99570 

11002 

99393 

12735 

99186 

14464 

98948 

16189 

98681 

41 

20 

09295 

99567 

11031 

99390 

12764 

99182 

14493 

98944 

16218 

98676 

40 

21 

09324 

99564 

11060 

99386 

12793 

99178  ! 

14522 

98940 

16246 

98671 

39 

22 

09353 

99562 

11089 

99383 

12822 

99175  i 

14551 

98936 

16275 

98667 

38 

23 

09382 

99559 

11118 

99380 

12851 

99171 

14580 

98931 

16304 

98662 

37 

24 

09411 

99556 

11147 

99377 

12880 

99167 

14608 

98927 

16333 

98657 

36 

25 

09440 

99553 

11176 

99374 

12908 

99163 

14637 

98923 

16361 

98652 

35 

26 

09469 

99551 

11205 

99370 

12937 

99160 

14666 

98919 

16390 

98648 

34 

27 

09498 

99548 

11234 

99367 

12966 

99156 

14695 

98914 

16419 

98643 

33 

28 

09527 

99545 

11263 

99364 

12995 

99152 

14723 

98910 

16447 

98638 

32 

29 

09556 

99542 

11291 

99360 

13024 

99148  i 

14752 

98906 

16476 

98633 

31 

30 

09585 

99540 

11320 

99357 

13053 

99144 

14781 

98902 

16505 

98629 

30 

31 

09614 

99537 

11349 

99354 

13081 

99141 

14810 

98897 

16533 

98624 

29 

32 

09642 

99534 

11378 

99351 

13110 

99137 

14838 

98893 

16562 

98619 

28 

33 

09671 

99531 

11407 

99347 

13139 

99133 

14867 

98889 

16591 

98614 

27 

34 

09700 

99528 

11436 

99344 

13168 

99129 

14896 

98884 

16620 

98609 

26 

35 

09729 

99526 

11465 

99341  ! 

13197 

99125 

14925 

98880 

16648 

98604 

25 

36 

09758 

99523 

11494 

99337  t 

13226 

99122 

14954 

98876 

16677 

98600 

24 

37 

09787 

99520 

11523 

99334 

13254 

99118 

14982 

98871 

16706 

98595 

23 

38 

09816 

99517 

11552 

99331 

13283 

99114 

15011 

98867 

16734 

98590 

22 

39 

09845 

99514 

11580 

99327 

13312 

99110 

15040 

98863 

16763 

98585 

21 

40 

09874 

99511 

11609 

99324 

13341 

99106 

15069 

98858 

16792 

98580 

20 

41 

09903 

99508 

11638 

99320 

13370 

99102  | 

15097 

98854 

16820 

98575 

19 

42 

09932 

99506 

11667 

99317 

13399 

99098 

15126 

98849 

16849 

98570 

18 

43 

09961 

99503 

11696 

99314 

13427 

99094 

15155 

98845 

16878 

98565 

17 

44 

09990 

99500 

11725 

99310 

13456 

99091 

15184 

98841 

16906 

98561 

16 

45 

10019 

99497 

11754 

99307 

13485 

99087 

15212 

98836 

16935 

98556 

15 

46 

10048 

99494 

11783 

99303 

13514 

99083 

15241 

98832 

16964 

98551 

14 

47 

10077 

99491 

11812 

99300  i 

13543 

99079 

15270 

98827 

16992 

98546 

13 

48 

10106 

99488 

11840 

99297 

13572 

99075 

15299 

98823 

17021 

98541 

12 

49 

10135 

99485 

11869 

99293 

13600 

99071  ! 

15327 

98818 

17050 

98536 

11 

50 

10164 

99482 

11898 

99290 

13629 

99067 

15356 

98814 

17078 

98531 

10 

51 

10192 

99479 

11927 

99286 

13658 

99063 

15385 

98809 

17107 

98526 

9 

52 

10221 

99476 

11956 

99283 

13687 

99059 

15414 

98805 

17136 

98521 

8 

53 

10250 

99473 

11985 

99279 

13716 

99055 

15442 

98800 

17164 

98516 

7 

54 

10279 

99470 

12014 

99276 

13744 

99051 

15471 

98796 

17193 

98511 

6 

55 

10308 

99467 

12043 

99272 

13773 

99047 

15500 

98791 

17222 

98506 

5 

56 

10337 

99464 

12071 

99269 

13802 

99043 

15529 

98787 

17250 

98501 

4 

57 

10366 

99461 

12100 

99265 

13831 

99039  | 

15557 

98782 

17279 

98496 

3 

58 

10395 

99458 

12129 

99262 

13860 

99035  ! 

15586 

98778 

17308 

98491 

2 

59 

10424 

99455 

12158 

99258 

13889 

99031 

15615 

98773 

17336 

98486 

1 

60 

10453 

99452 

12187 

99255 

13917 

99027 

15643 

98769 

17365 

98481 

0 

t 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

8^ 

1° 

& 

i° 

& 

3° 

81 

L° 

8< 

>° 

/ 

712 


APPENDIX. 


NATURAL.     SINES     AND    COSINES. 


10° 

11° 

13° 

13° 

14° 

/ 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine.    Sine. 

Cosine. 

r 

0 

17365 

98481 

19081 

98163 

20791 

97815 

22495 

97437 

24192 

97030 

60 

I 

17393 

98476 

19109 

98157 

20820 

97809 

22523 

97430 

24220 

97023 

59 

2 

17422 

98471  ! 

19138 

98152 

20848 

97803 

22552 

97424  , 

24249 

97015 

58 

3 

17451 

98466  i 

19167 

98146 

20877 

97797 

22580 

97417 

24277 

97008 

57 

4 

17479 

98461 

19195 

98140 

20905 

97791 

22608 

97411 

24305 

97001 

56 

5 

17508 

98455 

19224 

98135 

20933 

97784 

22637 

97404 

24333 

96994 

55 

6 

17537 

98450 

19252 

98129 

20962 

97778 

22665 

97398 

24362 

96987 

54 

7 

17565 

98445 

19281 

98124 

20990 

97772 

22693 

97391 

24390 

96980 

53 

8 

17594 

98440 

19309 

98118 

21019 

97766 

22722 

97384 

24418 

96973 

52 

9 

17623 

98435 

19338 

98112 

21047 

97760 

22750 

97378 

24446 

96966 

51 

10 

17651 

98430 

19366 

98107 

21076 

97754 

22778 

97371 

24474 

96959 

50 

11 

17680 

98425 

19395 

98101 

21104 

97748 

22807 

97365 

24503 

96952 

49 

12 

17708 

98420 

19423 

98096 

21132 

97742 

22835 

97358 

24531 

96945 

48 

13 

17737 

98414 

19452 

98090 

21161 

97735 

22863 

97351 

24559 

96937 

47 

14 

17766 

98409 

19481 

98084 

21189 

97729 

22892 

97345 

24587 

96930 

46 

15 

17794 

98404 

19509 

98079 

21218 

97723 

22920 

97338 

24615 

96923 

45 

16 

17823 

98399 

19538 

98073 

21246 

97717 

22948 

97331 

24644 

96916 

44 

17 

17852 

98394 

19566 

98067 

21275 

97711 

22977 

97325 

24672 

96909 

43 

18 

17880 

98389 

19595 

98061 

21303 

97705 

23005 

97318 

24700 

96902 

42 

19 

17909 

98383 

19623 

98056 

21331 

97698 

23033 

97311 

24728 

96894 

41 

20 

17937 

98378 

19652 

98050 

21360 

97692 

'  23062 

97304   24756 

96887 

40 

21 

17966 

98373 

19680 

98044 

21388 

97686 

:  23090 

97298   24784 

96880 

39 

22 

17995 

98368 

19709 

98039 

21417 

97680 

23118 

97291   24813 

96873 

38 

23 

18023 

98362 

19737 

98033 

21445 

97673 

23146 

97284   24841 

96866 

37 

24 

18052 

98357 

19766 

98027 

21474 

97667 

23175 

97278   24869 

96858 

36 

25 

18081 

98352 

19794 

98021 

21502 

97661 

23203 

97271  :  24897 

96851 

35 

26 

18109 

98347 

19823 

98016 

21530 

97655 

23231 

97264 

24925 

96844 

34 

27 

18138 

98341 

19851 

98010 

21559 

97648 

23260 

97257 

24954 

96837 

33 

28 

18166 

98336 

19880 

98004 

21587 

97642 

23288 

P7251 

24982 

96829 

32 

29 

18195 

98331 

19908 

97998 

21616 

97636 

23316 

97244 

25010 

96822 

31 

30 

18224 

98325 

19937 

97992 

21644 

97630 

23345 

97237 

25038 

96815 

30 

31 

18252 

98320 

19965 

97987 

21672 

97623 

23373 

97230 

25066 

96807 

29 

32 

18281 

98315 

19994 

97981 

21701 

97617 

23401 

97223 

25094 

96800 

28 

33 

18309 

98310 

20022 

97975 

21729 

97611 

23429 

97217 

25122 

96793 

27 

34 

18338 

98304 

20051 

97969 

21758 

97604 

23458 

97210 

25151 

96786 

26 

35 

18367 

98299 

20079 

97963 

21786 

97598 

23486 

97203 

25179 

96778 

25 

36 

18395 

98294 

20108 

97958 

21814 

97592 

23514 

97196 

25207 

96771 

24 

37 

18424 

98288 

20136 

97952 

21843 

97585 

23542 

97189 

25235 

96764 

23 

38 

18452 

98283 

20165 

97946 

21871 

97579 

23571 

97182 

25263 

96756 

22 

39 

18481 

98277 

20193 

97940 

21899 

97573 

23599 

97176 

25291 

96749 

21 

40 

18509 

98272 

20222 

97934 

21928 

97566 

23627 

97169 

25320 

96742 

20 

41 

18538 

98267 

20250 

97928 

21956 

97560 

23656 

97162 

25348 

96734 

19 

42 

18567 

98261 

20279 

97922 

21985 

97553 

23684 

97155 

25376 

96727 

18 

43 

18595 

98256 

20307 

97916 

22013 

97547 

23712 

97148 

25404 

96719 

17 

44 

18624 

98250 

20336 

97910 

22041 

97541 

23740 

97141 

25432 

96712 

16 

45 

18652 

98245 

20364 

97905 

22070 

97534 

23769 

97134 

25460 

96705 

15 

46 

18681 

98240 

20393 

97899 

22098 

97528 

23797 

97127 

25488 

96697 

14 

47 

18710 

98234 

20421 

97893 

22126 

97521 

23825 

97120 

25516 

96690 

13 

48 

18738 

98229 

20450 

97887 

22155 

97515  i  23853 

97113 

25545 

96682 

12 

49 

18767 

98223 

20478 

97881 

22183 

97508 

23882 

97106 

25573 

96675 

11 

50 

18795 

98218 

20507 

97875 

22212 

97502 

23910 

97100 

25601 

96667 

10 

51 

18824 

98212 

20535 

97869  1  22240 

97496 

23938 

97093 

25629 

96660 

9 

52 

18852 

98207 

20563 

97863  i  22268 

97489 

23966 

97086 

25657 

96653 

8 

53 

18881 

98201 

20592 

97857   22297 

97483 

23995 

97079 

25685 

96645 

7 

54 

18910 

98196 

20620 

97851 

22325 

97476 

24023 

97072 

25713 

96638 

6 

55 

18938 

98190 

20649 

97845 

±_>:55:j 

97470 

24051 

97065 

25741 

96630 

5 

56 

18967 

98185   20677 

97839 

22382 

97463 

24079 

97058 

25769 

96623 

4 

57 

18995 

98179 

20706 

97833 

22410 

97457 

24108 

97051 

25798 

96615 

3 

58 

19024 

98174 

20734 

97827 

22438 

97450 

24136 

97044 

25826 

96608 

2 

59 

19052 

98168 

20763 

97821 

22467 

97444 

24164 

97037 

25854 

96600 

1 

60 

19081 

98163 

20791 

97815   22495 

97437 

24192 

97030 

25882 

96593 

0 

Cosine. 

Sin,.. 

Cosine. 

Sine.    Cosine. 

Sine.    Cosine. 

Sine. 

Cosine. 

Sine. 

/ 

7O° 

T8o          77-0     Ij     76° 

75° 

/ 

APPENDIX. 


713 


NATURAL,     SINKS     AND    COSINES. 


15° 

ie° 

17° 

18° 

1O° 

r 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

/ 

0 

25882 

96593 

27564 

96126 

29237 

95630 

30902 

95106 

32557 

94552 

60 

1 

25910 

96585 

27592 

96118  ||  29265 

95622 

30929 

95097 

32584 

94542 

59 

2 

25938 

96578 

27620 

96110   29293 

95613 

30957 

95088 

32612 

94533 

58 

3 

25966 

96570 

27648 

96102 

29321 

95605 

30985 

95079 

32639 

94523 

57 

4 

25994 

96562 

27676 

96094 

29348 

95596 

,  31012 

95070 

32667 

94514 

56 

5 

26022 

96555 

27704 

96086 

29376 

95588 

31040 

95061  |  32694 

94504 

55 

6 

26050 

96547 

27731 

96078 

29404 

95579 

31068 

95052 

32722 

94495 

54 

7 

26079 

96540 

27759 

96070 

29432 

95571 

31095 

95043 

32749 

94485 

53 

8 

26107 

96532 

27787 

96062 

29460 

95562 

31123 

95033 

32777 

94476 

52 

9 

26135 

96524 

27815 

96054 

29487 

95554 

31151 

95024 

32804 

94466 

51 

10 

26163 

96517 

27843 

96046 

29515 

95545 

31178 

95015 

32832 

94457 

50 

11 

26191 

96509 

27871 

96037 

29543 

95536 

31206 

95006 

32859 

94447 

49 

12 

26219 

96502 

27899 

96029 

29571 

95528 

1  31233 

94997 

32887 

94438 

48 

13 

26247 

96494 

27927 

96021 

29599 

95519  1  31261 

94988  1  32914 

94428 

47 

14 

26275 

96486 

27955 

96013 

29626 

95511   31289 

94979  |j  32942 

94418 

46 

15 

26303 

96479 

27983 

96005 

29654 

95502 

31316 

94970   32969 

94409 

45 

16 

26331 

96471 

28011 

95997 

29682 

95493 

31344 

94961   32997 

94399 

44 

17 

26359 

96463 

28039 

95989 

29710 

95485 

31372 

94952  |!  33024 

94390 

43 

18 

26387 

96456 

28067 

95981 

29737 

95476  ;  31399 

94943 

33051 

94380 

42 

19 

26415 

96448 

28095 

95972 

29765 

95467  :  31427 

94933 

33079 

94370 

41 

20 

26443 

96440 

28123 

95964 

29793 

95459  j;  31454 

94924 

33106 

94361 

40 

21 

26471 

96433 

28150 

95956 

29821 

95450 

I  31482 

94915 

33134 

94351 

39 

22 

26500 

9tf425  I  28178 

95948 

29849 

95441 

31510 

94906 

33161 

94342   38 

23 

26528 

96417   28206 

95940  ! 

29876 

95433 

31537 

94897 

33189 

94332   37 

24 

26556 

96410  |  28234 

95931 

29904 

95424  i  31565 

94888 

33216 

94322   36 

25 

26584 

96402  l\  28262 

95923 

29932 

95415  '  31593 

94878 

33244 

94313   35 

26 

26612 

96394  !i  28290 

95915 

29960 

95407   31620 

94869 

33271 

94303 

34 

27 

26640 

96386  !  28318 

95907 

29987 

95398   31648 

94860 

33298 

94293 

33 

28 

26668 

96379  j  28346 

95898 

30015 

95389   31675 

94851 

33326 

94284 

32 

29 

26696 

96371  |  28374 

95890 

30043 

95380 

31703 

94842 

33353 

94274 

31 

30 

26724 

96363  j  28402 

95882 

30071 

95372  1  31730 

94832 

33381 

94264 

30 

31 

26752 

96355   28429 

95874 

30098 

"95363  !  31758 

94823 

33408 

94254 

29 

32 

26780 

96347   28457 

95865 

30126 

95354   31786 

94814 

33436 

94245 

28 

33 

26808 

96340   28485 

95857 

30154 

95345  :i  31813 

94805 

33463 

94235 

27 

34 

26836 

96332   28513 

95849 

30182 

95337  i  31841 

94795 

33490 

94225 

26 

35 

26864 

96324  ;  28541 

95841 

30209 

95328   31868 

94786 

33518 

94215 

25 

36 

26892 

96316   28569 

95832 

30237 

95319   31896 

94777 

33545 

94206 

24 

37 

26920 

96308   28597 

95824  l|  30265 

95310   31923 

94768 

33573 

94196 

23 

38 

26948 

96301   28625 

95816 

30292 

95301   31951 

94758 

33600 

94186 

22 

39 

26976 

96293 

!  28652 

95807 

30320 

95293   31979 

94749 

33627 

94176 

21 

40 

27004 

96285 

28680 

95799 

30348 

95284 

32006 

94740 

33655 

94167 

20 

41 

27032 

96277 

28708 

95791 

30376 

95275 

32034 

94730 

33682 

94157 

19 

42 

27060 

96269 

28736 

95782  i  30403 

95266 

32061 

94721 

33710 

94147 

18 

43 

27088 

96261 

28764 

95774 

30431 

95257 

32089 

94712 

33737 

94137 

17 

44 

27116 

96253 

28792 

95766 

30459 

95248 

32116 

94702 

33764 

94127 

16 

45 

27144 

96246 

28820 

95757 

30486 

95240 

32144 

94693 

33792 

94118 

15 

46 

27172 

96238 

28847 

95749 

30514 

95231 

32171 

94684 

33819 

94108 

14 

47 

27200 

96230 

!  28875 

95740 

30542 

95222 

'  32199 

94674 

33846 

94098 

13 

48 

27228 

96222  !  28903 

95732 

30570 

95213 

32227 

94665 

33874 

94088 

12 

49 

27256 

96214  >  28931 

95724 

30597 

95204  ||  32254 

94656 

33901 

94078 

11 

50 

27284 

96206 

28959 

95715 

30625 

95195  :  32282 

946-46 

33929 

94068 

10 

51 

27312 

96198 

28987 

95707 

30653 

95186  jj  32309 

94637 

33956 

94058 

9 

52 

27340 

96190   29015 

95698 

30680 

95177   32337 

94627 

33983 

94049 

8 

53 

27368 

96182   29042 

95690 

30708 

95168   32364 

94618 

34011 

94039 

7 

54 

27396 

96174  j  29070 

95681  |  30736 

95159   32392 

94609  i  34038 

94029 

6 

55 

27424 

96166  |  29098 

95673   30763 

95150  •  32419 

94599   34065 

94019 

5 

56 

27452 

96158   29126 

95664   30791 

95142   32447 

94590  i  34093 

94009   4 

57 

27480 

96150   29154 

95656  I!  30819 

95133   32474 

94580   34120 

93999 

3 

58 

27508 

96142   29182 

95647  :'  30846 

95124   32502 

94571 

34147 

93989 

2 

59 

27536 

96134 

29209 

95639   30874 

95115 

i  32529 

94561 

34175 

93979 

1 

60 

27564 

96126 

29237 

95630   30902 

95106 

32557 

94552 

34202 

93969 

0 

Cosine. 

Sine. 

Cosine. 

Sine.    Cosine. 

Sine.    Cosine. 

Sine. 

Cosine. 

Sine. 

/ 

7-4,0 

7-30      |     730     ||     T-XO 

7O° 

/ 

714: 


APPENDIX. 


NATURAL,     SINES     AND    COSINES. 


30° 

31° 

33° 

33° 

34° 

/ 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

/ 

0 

34202 

93969 

35837 

93358 

37461 

92718 

39073 

92050 

40674 

91355 

60 

1 

34229 

93959 

35864 

93348 

37488 

92707 

39100 

92039 

40700 

91343 

59 

2 

34257 

93949 

35891 

93337 

37515 

92697 

39127 

92028 

40727 

91331 

58 

3 

342S4 

93939 

35918 

93327 

37542 

92686 

39153 

92016 

40753 

91319 

57 

4 

34311 

93929 

35945 

93316 

37569 

92675 

39180 

92005 

40780 

91307 

56 

5 

34339 

93919 

35973 

93306 

37595 

92664 

39207 

91994 

40806 

91295 

55 

6 

34366 

93909 

36000 

93295 

37t>22 

92653 

39234 

91982 

40833 

91283 

54 

7 

34393 

93899 

36027 

93285 

37649 

92642 

39260 

91971 

40860 

91272 

53 

8   34421 

93889 

36054 

93274 

37676 

92631 

39287 

91959 

40886 

91260 

52 

9 

34448 

93879 

36081 

93264 

37703 

92620 

39314 

91948 

40913 

91248 

51 

10 

34475 

93869 

36108 

93253 

37730 

92609 

39341 

91936 

40939 

91236 

50 

11 

34503 

93859 

36135 

93243 

37757 

92598 

39367 

91925 

40966 

91224 

49 

12 

34530 

93849 

36162 

93232 

37784 

92587 

39394 

91914 

40992 

91212 

48 

13 

34557 

93839 

36190 

93222 

37811 

92576 

39421 

91902 

41019 

91200 

47 

14 

34584 

93829 

36217 

93211 

37838 

92565 

39448 

91891 

41045 

91188 

46 

15 

34612 

93819 

36244 

93201 

37865 

92554 

39474 

91879 

41072 

91176 

45 

16 

34639 

93809 

36271 

93190 

37892 

92543 

39501 

91868 

41098 

91164 

44 

17 

34666 

93799 

36298 

93180 

37919 

92532 

39528 

91856 

41125 

91152 

43 

18 

34694 

93789 

36325 

93169 

i  37946 

92521 

39555 

91845 

41151 

91140 

42 

19 

34721 

93779 

36352 

93159 

37973 

92510 

39581 

91833 

41178 

91128 

41 

20 

34748 

93769 

36379 

93148 

37999 

92499 

39608 

91822 

41204 

91116 

40 

21 

34775 

93759 

36406 

93137 

38026 

92488 

39635 

91810 

41231 

91104 

39 

22 

34803 

93748 

36434 

93127 

38053 

92477 

39661 

91799 

41257 

91092 

38 

23 

34830 

93738 

36461 

93116 

38080 

92466 

39688 

91787 

41284 

91080 

37 

24 

34857 

93728 

36488 

93106 

38107 

92455 

39715 

91775 

41310 

91068 

36 

25 

34884 

93718 

36515 

93095 

38134 

92444 

39741 

91764 

41337 

91056 

35 

26 

34912 

93708 

36542 

93084 

38161 

92432 

39768 

91752 

41363 

91044 

34 

27 

34939 

93698 

36569 

93074 

38188 

92421 

39795 

91741 

41390 

91032 

33 

28 

34966 

93688 

36596 

93063 

38215 

92410 

39822 

91729 

41416 

91020 

32 

29 

34993 

93677 

36623 

93052 

38241 

92399 

39848 

91718 

41443 

91008 

31 

30 

35021 

93667 

36650 

93042 

38268 

92388 

39875 

91706 

41469 

90996 

30 

31 

35048 

93657 

36677 

93031 

38295 

92377 

39902 

91694 

41496 

90984 

29 

32 

35075 

93647 

36704 

93020 

38322 

92366 

39928 

91683 

41522 

90972 

28 

33 

35102 

93637 

36731 

93010 

38349 

92355 

39955 

91671 

41549 

90960 

27 

34 

35130 

93626 

36758 

92999 

38376 

92343 

39982 

•  91660 

41575 

90948 

26 

35 

35157 

93616 

36785 

92988 

38403 

92332 

40008 

91648 

!  41602 

90936 

25 

36 

35184 

93606 

36812 

92978 

38430 

92321 

40035 

91636 

]  41628 

90924 

24 

37 

35211 

93596 

36839 

92967 

38456 

92310 

40062 

91625 

41  655 

90911 

23 

38 

35239 

93585 

36867 

92956 

38483 

92299 

40088 

91613 

41681 

90899 

22 

39 

35266 

93575 

36894 

92945 

38510 

92287 

40115 

91601 

41707 

90887 

21 

40 

35293 

93565 

36921 

92935 

38537 

92276 

40141 

91590 

41734 

90875 

20 

41 

35320 

93555 

36948 

92924 

38564 

92265 

40168 

91578 

41760 

90863 

19 

42 

35347 

93544 

36975 

92933 

38591 

92254 

40195 

91566 

41787 

90851 

18 

43 

35375 

93534 

37002 

92902 

38617 

92243 

40221 

91555 

41813 

90839 

17 

44 

35402 

93524 

37029 

92892 

38644 

92231  j  40248 

91543 

41840 

90826 

16 

45 

35429 

93514 

37056 

92881 

38671 

92220 

40275 

91531 

41866 

90814 

15 

46 

35456 

93503 

37083 

92870 

38698 

92209 

40301 

91519 

41892 

90802 

14 

47 

35484 

93493 

37110 

92859   38725 

92198 

40328 

91508 

41919 

90790   13 

48 

35511 

93483 

37137 

92849   38752 

92186   40355 

91496 

41945 

90778   1  -1 

49 

35538 

93472 

37164 

92838  1  38778 

92175 

40381 

91484 

41972 

90766 

11 

60 

35565 

93462 

37191 

92827  1  38805 

92164 

40408 

91472 

41998 

90753 

10 

51 

35592 

93452 

37218 

92816 

38832 

92152 

40434 

91461 

42024 

90741 

9 

52 

35619 

93441 

37245 

92805 

38859 

92141 

40461 

91449 

42051 

90729 

8 

53 

35647 

93431 

37272 

92794 

38886 

92130 

40488 

91437 

42077 

90717 

7 

54 

35674 

93420 

37299 

92784 

38912 

92119 

40514 

91425 

42104 

90704 

6 

55 

35701 

93410 

37326 

92773  i  38939 

92107 

40541 

91414 

42130 

90692 

5 

56 

35728 

93400 

37353 

92762  1  38966 

92096 

40567 

91402 

42156 

90680 

4 

57 

35755 

93389 

37380 

92751 

38993 

92085 

40594 

91390 

42183 

90668 

3 

58 

35782 

93379 

37407 

92740 

39020 

92073 

40621 

91378 

42209 

90655 

2 

59 

35810 

93368 

37434 

92729 

39046 

92062 

40647 

91366 

42235 

90643 

1 

60 

35837 

93358 

37461 

92718 

39073 

92050 

40674 

91355 

42262 

90631 

0 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Stoe. 

Cosine. 

Sine. 

Cosine. 

Sine. 

/ 

eo° 

68° 

G7° 

GG° 

015° 

/ 

APPENDIX. 


715 


NATURAL.     SINES     AND    COSINES. 


35° 

30° 

37°           38°           39° 

/ 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

/ 

0 

42262 

90631  i  43837 

89879 

45399 

89101 

46947 

88295 

48481 

87462 

60 

I 

42288 

90618  '  43863 

89867 

45425 

89087  ! 

46973 

88281 

48506 

87448 

59 

2 

42315 

90606  :  43889 

89854 

45451 

89074 

46999 

88267 

48532 

87434  !  58 

3 

42341 

90594   43916 

89841 

45477 

89061 

47024 

88254  j  48557 

87420 

57 

4 

42367 

90582   43942 

89828 

45503 

89048 

47050 

88240 

48583 

87406 

56 

5   42394 

90569  j  43968 

89816 

45529 

89035 

47076 

88226 

48608 

87391   55 

6  !  42420 

90557  !  43994 

89803 

45554 

89021 

47101 

88213 

48634 

87377   54 

7 

42446 

90545 

44020 

89790 

45580 

89008 

47127 

88199 

48659 

87363 

53 

8 

42473 

90532 

44046 

89777 

45606 

88995 

47153 

88185 

48684 

87349 

52 

9 

42499 

90520 

44072 

89764 

45632 

88981 

47178 

88172 

48710 

87335   51 

10 

42525 

90507 

44098 

89752 

45658 

88968 

47204 

88158 

48735 

87321   50 

11 

42552 

90495 

44124 

89739 

45684 

88955 

47229 

88144 

48761 

87306   49 

12 

42578 

90483 

44151 

89726 

45710 

88942  ; 

•47255 

88130 

48786 

87292   48 

13 

42604 

90470 

44177 

89713 

45736 

88928 

47281 

88117 

48811 

87278  47 

14 

42631 

90458 

44203 

89700 

45762 

88915 

47306 

88103 

48837 

87264 

46 

15 

42657 

90446 

44229 

89687 

45787 

88902 

47332 

88089 

48862 

87250 

45 

16 

42683 

90433 

44255 

89674 

45813 

88888 

47358 

88075 

48888 

87235 

44 

17 

42709 

90421 

44281 

89662 

45839 

88875  : 

47383 

88062  !  48913 

87221 

43 

18 

42736 

90408 

44307 

89649 

45865 

88862   47409 

88048  !  48938 

87207 

42 

19 

42762- 

90396 

44333 

89636  !  45891 

88848   47434 

88034  :j  48964 

87193 

41 

20 

42788 

90383 

44359 

89623 

45917 

88835  i  47460 

88020  i!  48989 

87178 

40 

21 

42815 

90371 

44385 

89610 

45942 

88822  i  47486 

88006  !  49014 

87164 

39 

22 

42841 

90358 

44411 

89597 

45968 

88808   47511 

87993  !  49040 

87150 

38 

23 

42867 

90346 

44437 

89584 

45994 

88795   47537 

87979  I  49065 

87136 

37 

24 

42894 

90334 

44464 

89571 

46020 

88782 

47562 

87965  -i  49090 

87121 

36 

25 

42920 

90321  ; 

44490 

89558  ! 

46046 

88768 

47588 

87951  ; 

49116 

87107 

35 

26 

42946 

90309 

44516 

89545  1  46072 

88755  : 

47614 

87937 

49141 

87093 

34 

27 

42972 

90296 

44542 

89532  :!  46097 

88741  |  47639 

87923 

49166 

87079   33 

28 

42999 

90284 

44568 

89519   46123 

88728 

47665 

87909 

49192 

87064   32 

29 

43025 

90271 

44594 

89506  :  46149 

88715  : 

47690 

87896 

49217 

87050   31 

30 

43051 

90259 

44620 

89493  j  46175 

88701 

47716 

87882  jj  49242 

87036  |  30 

31 

43077 

90246 

44646 

89480  II  46201 

88688 

47741 

87868  !  49268 

87021 

29 

32 

43104 

90233 

44672 

89467  : 

46226 

88674 

47767 

87854  i  49293 

87007   28 

33 

43130 

90221 

44698 

89454 

46252 

88661 

47793 

87840  |  49318 

86993   27 

34 

43156 

90208 

44724 

89441 

46278 

88647 

47818 

87826  jl  49344 

86978   26 

35 

43182 

90196 

44750 

89428 

46304 

88634 

47844 

87812  !  49369 

86964  !  25 

36 

43209 

90183 

44776 

89415 

46330 

88620 

47869 

87798  i  49394 

86949   24 

37 

43235 

90171  ! 

44802 

89402 

46355 

88607 

47895 

87784  1  49419 

86935 

23 

38 

43261 

90158  i 

44828 

89389  ; 

46381 

88698   47920 

87770  ! 

49445 

86921 

22 

39 

43287 

90146 

44854 

89376 

46407 

88580   47946 

87756  i 

49470 

86906 

21 

40 

43313 

90133 

44880 

89363  ! 

46433 

88566  !  47971 

87743 

49495 

86892 

20 

41 

43340 

90120 

44906 

89350   46458 

88553   47997 

87729 

49521 

86878 

19 

42 

43366 

90108 

44932 

89337   46484 

88539   48022 

87715  ! 

49546 

86863 

18 

43 

43392 

90095 

44958 

89324 

46510 

88526 

48048 

87701 

49571 

86849 

17 

44 

43418 

90082 

44984 

89311 

46536 

88512 

48073 

87687 

49596 

86834 

16 

45 

43445 

90070 

45010 

89298 

46561 

88499 

48099 

87673 

49622 

86820 

15 

46 

43471 

90057  I  45036 

89285 

46587 

88485 

48124 

87659  i 

49647 

86805 

14 

47 

43497 

90045 

45062 

89272   46613 

88472  '• 

48150 

87645 

49672 

86791 

13 

48 

43523 

90032  ; 

45088 

89259 

46639 

88458 

48175 

87631 

49697 

86777 

12 

49 

43549 

90019 

45114 

89245 

46664 

88445 

48201 

87617 

49723 

86762 

11 

50 

43575 

90007 

45140 

89232 

46690 

88431  i 

48226 

87603 

49748 

86748 

10 

51 

43602 

89994 

45166 

89219 

46716 

88417 

48252 

87589 

49773 

86733 

9 

52 

43628 

89981 

45192 

89206 

46742 

88404 

48277 

87575 

49798 

86719 

8 

53 

43654 

89968 

45218 

89193 

46767 

88390 

48303 

87561 

49824 

86704 

7 

54 

43680 

89956 

45243 

89180 

46793 

88377  i 

48328 

87546 

49849 

86690 

6 

55 

43706 

89943  ! 

45269 

89167 

46819 

88363 

48354 

87532 

49874 

86675 

5 

56 

43733 

89930 

45295 

89153  ! 

46844 

88349  j 

48379 

87518 

49899 

86661 

4 

57 

43759 

89918 

45321 

89140 

46870 

88336 

48405 

87504 

49924 

86646 

3 

58 

43785 

89905 

45347 

89127 

46896 

88322 

48430 

87490 

49950 

86632 

2 

59 

43811 

89892 

45373 

89114 

46921 

88308 

48456 

87476 

49975 

86617 

1 

60 

43837 

89879 

45399 

89101 

46947 

88295 

48481 

87462 

50000 

86603 

0 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine.  I 

Cosine. 

Sine. 

/ 

G4,o 

G3°          63° 

01° 

G0° 

f 

716 


APPENDIX. 


NATURAL,     SINES     AND    COSINES. 


30° 

31° 

33° 

33° 

34° 

' 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

7 

0 

50000 

86603 

51504 

85717 

52992 

84805 

54464 

83867 

55919 

82904 

60 

1 

50025 

86588 

51529 

85702 

53017 

84789 

54488 

83851 

55943 

82887 

59 

2 

50050 

86573 

51554 

85687 

53041 

84774 

54513 

83835 

55968 

82871 

58 

3 

50076 

86559 

51579 

85672 

53066 

84759 

54537 

83819 

55992 

82855 

57 

4 

50101 

86544 

51604 

85657 

53091 

84743 

54561 

83804  i  56016 

82839 

56 

5 

50126 

86530 

51628 

85642 

53115 

84728 

54586 

83788  :  56040 

82822 

55 

6 

50151 

86515 

51653 

85627 

53140 

84712 

54610 

83772  j  56064 

82806 

54 

7 

50176 

86501 

51678 

85612 

53164 

84697 

54635 

83756  i  56088 

82790 

53 

8 

50201 

86486 

51703 

85597 

53189 

84681 

54659 

83740  1  56112 

82773 

52 

9 

50227 

86471 

51728 

85582 

53214 

84666 

54683 

83724  ;  56136 

82757 

51 

10 

50252 

86457 

51753 

85567 

53238 

84650 

54708 

83708  '  56160 

82741 

50 

11 

50277 

86442 

51778 

85551 

53263 

84635 

54732 

83692 

56184 

82724 

49 

12 

50302 

86427 

51803 

85536 

53288 

84619 

54756 

83676 

56208 

82708 

48 

13 

50327 

86413 

51828 

85521 

53312 

84604 

54781 

83660   56232 

82692 

47 

14 

50352 

86398 

51852 

85506 

53337 

84588 

54805 

83645 

56256 

82675 

46 

15 

50377 

86384 

51877 

85491 

53361 

84573 

54829 

83629 

56280 

82659 

45 

16 

50403 

86369 

51902 

85476 

53386 

84557 

•  54854 

83613 

56305 

82643 

44 

17 

50428 

86354 

51927 

85461 

53411 

84542 

54878 

83597  1  56329 

82626 

43 

18 

50453 

86340 

51952 

85446  i 

53435 

84526 

54902 

83581  i  56353 

82610 

42 

19 

50478 

86325 

51977 

85431 

53460 

84511 

54927 

83565   56377 

82593 

41 

20 

50503 

86310 

52002 

85416 

53484 

84495 

54951 

83549 

56401 

82577 

40 

21 

50528 

86295 

52026 

85401 

53509 

84480  i  54975 

83533 

56425 

82561 

39 

22 

50553 

86281 

52051 

85385 

53534 

84464  ;  54999 

83517 

56449 

82544 

38 

23 

50578 

86266 

52076 

85370 

53558 

84448   55024 

83501 

56473 

82528 

37 

24 

50603 

86251 

52101 

85355 

53583 

84433  h  55048 

83485 

56497 

82S11 

36 

25 

50628 

86237 

52126 

85340 

53607 

84417 

55072 

83469 

56521 

82495 

35 

26 

50654 

86222 

52151 

85325 

53632 

84402 

55097 

83453 

56545 

82478 

34 

27 

50679 

86207 

52175 

85310 

53656 

84386 

55121 

83437 

56569 

82462 

33 

28 

50704 

86192 

52200 

85294  j 

53681 

84370 

55145 

83421 

56593 

82446 

32 

29 

50729 

86178 

52225 

85279 

53705 

84355 

55169 

83405 

56617 

82429 

31 

30 

50754 

86163 

52250 

85264 

53730 

84339 

55194 

83389 

56641 

82413 

30 

31 

50779 

86148 

52275 

85249 

53754 

84324 

55218 

83373 

56665 

82396 

29 

32 

50804 

86133 

52299 

85234 

53779 

84308 

55242 

83356 

\  56689 

82380 

28 

33 

50829 

86119 

52324 

85218 

53804 

84292 

55266 

83340 

56713 

82363 

27 

34 

50854 

86104 

52349 

85203 

53828 

84277 

55291 

83324 

56736 

82347 

26 

35 

50879 

86089,!  52374 

85188 

53853 

84261 

55315 

83308 

56760 

82330 

25 

36 

50904 

86074  I  52399 

85173  ! 

53877 

84245 

55339 

83292 

56784 

82314 

24 

37 

50929 

86059   52423 

85157 

53902 

84230 

55363 

83276 

56808 

82297 

23 

38 

50954 

86045   52448 

85142  ! 

53926 

84214 

55388 

83260 

:  56832 

82281 

22 

39 

50979 

86030  1  52473 

85127 

53951 

84198 

55412 

83244 

56856 

82264 

21 

40 

51004 

86015 

52498 

85112 

53975 

84182 

55436 

83228 

56880 

82248 

20 

41 

51029 

86000 

52522 

85096 

54000 

84167 

55460 

83212 

56904 

82231 

19 

42 

51054 

85985 

52547 

85081 

54024 

84151 

55484 

83195 

56928 

82214 

18 

43 

51079 

85970 

52572 

85066 

54049 

84135 

55509 

83179 

56952 

82198 

17 

44 

51104 

85956 

52597 

85051 

54073 

84120 

55533 

83163 

56976 

82181 

16 

45 

51129 

85941 

52621 

85035 

54097 

84104 

55557 

83147 

57000 

82165 

15 

46 

51154 

85926 

52646 

85020 

54122 

84088  1  55581 

83131 

57024 

82148 

14 

47 

51179 

85911 

52671 

85005 

54146 

84072   55605 

83115 

57047 

82132 

13 

48 

51204 

85896 

52696 

84989 

54171 

84057   55630 

83098 

57071 

82115 

12 

49 

51229 

85881 

52720 

84974 

54195 

84041 

55654 

83082 

57095 

82098 

11 

50 

51254 

85866 

52745 

84959 

54220 

84025 

55678 

83066 

57119 

82082 

10 

51 

51279 

85851 

52770 

84943 

54244 

84009 

55702 

83050 

57143 

82065 

9 

52 

51304 

85836 

52794 

84928 

54269 

83994 

55726 

83034 

57167 

82048 

8 

53 

51329 

85821 

52819 

84913 

54293 

83978 

55750 

83017 

57191 

82032 

7 

54 

51354 

85806 

52844 

84897 

54317 

83962 

55775 

83001 

57215 

82015 

6 

55 

51379 

85792 

52869 

84882 

54342 

83946 

55799 

82985 

57238 

81999 

5 

56 

51404 

85777 

52893 

84866 

54366 

83930   55823 

82969 

57262 

81982 

4 

57 

51429 

85762 

52918 

84851 

54391 

83915   55847 

82953 

57286 

81965 

3 

58 

51454 

85747 

52943 

84836 

54415 

83899 

55871 

82936 

57310 

81949 

2 

59 

51479 

85732 

52967 

84820 

54440 

83883 

55895 

82920 

57334 

81932 

1 

60 

51504 

85717 

52992 

84805 

54464 

83867 

55919 

82904 

57358 

81915 

0 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

f 

50° 

58° 

57-° 

56°     ij     55° 

APPENDIX. 


TIT 


NATURAL.     SINKS     AND    COSINES. 


3.5° 

3G° 

37° 

38° 

3O° 

/ 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

/ 

0 

57358 

81915 

58779 

80902 

60182 

79864 

61566 

78801 

62932 

77715 

60 

I 

57381 

81899 

58802 

80885 

60205 

79846 

61589 

78783 

62955 

77696 

59 

2 

57405 

81882 

58826 

80867   60228 

79829 

61612 

78765 

62977 

77678 

58 

3 

57429 

81865 

58849 

80850 

60251 

79811 

61635 

78747 

63000 

77660  1  57 

4 

57453 

81848 

58873 

80833 

60274 

79793 

61658 

78729 

63022 

77641 

56 

5 

57477 

81832 

58896 

80816 

60298 

79776 

61681 

78711 

63045 

77623 

55 

6 

57501 

81815 

58920 

80799 

60321 

79758 

61704 

78694 

63068 

77605 

54 

7 

57524 

81798 

58943 

80782   60344 

79741 

61726 

78676 

63090 

77586 

5S 

8 

57548 

81782 

58967 

80765   60367 

79723 

61749 

78658 

63113 

77568 

52 

9 

57572 

81765 

58990 

80748   60390 

79706 

61772 

78640 

63135 

77550 

51 

10 

57596 

81748 

59014 

80730  l  60414 

79688 

61795 

78622 

63158 

77531 

50 

11 

57619 

81731 

59037 

80713  :  60437 

79671 

61818 

78604 

63180 

77513 

49 

12 

57643 

81714 

59061 

80696   60460 

79653 

61841 

78586 

63203 

77494 

48 

13 

57667 

81698 

59084 

80679   60483 

79635 

61864 

78568 

63225 

77476 

47 

14 

57691 

81681 

59108 

80662   60506 

79618 

61887 

78550 

63248 

77458 

46- 

15 

57715 

81664 

59131 

80644  i  60529 

79600 

61909 

78532 

63271 

77439 

45 

16 

57738 

81647 

59154 

80627   60553 

79583 

61932 

78514 

63293 

77421 

44 

17 

57762 

81631 

59178 

80610   60576 

79565 

61955 

78496 

63316 

77402 

4S 

18 

57786 

81614 

59201 

80593   60599 

79547 

61978 

78478 

63338 

77384 

42 

19 

57810 

81597 

59225 

80576  !  60622 

79530 

62001 

78460 

63361 

77366 

41 

20 

57833 

81580 

59248 

80558   60645 

79512 

62024 

78442 

63383 

77347 

40 

21 

57857 

81563 

59272 

80541   60668 

79494 

62046 

78424 

63406 

77329 

39 

22 

57881 

81546 

59295 

80524  ;  60691 

79477 

62069 

78405 

63428 

77310 

38 

23 

57904 

81530 

59318 

80507  j  60714 

79459 

62092 

78387 

63451 

77292 

37 

24 

57928 

81513 

59342 

80489 

60738 

79441 

62115 

78369 

63473 

77273 

36- 

25 

57952 

81496 

59365 

80472 

60761 

79424 

62138 

78351 

63496 

77255 

35 

26 

57976 

81479 

59389 

80455 

60784 

79406 

62160 

78333 

63518 

77236   34 

27 

57999 

81462 

59412 

80438  ! 

60807 

79388 

62183 

78315 

63540 

77218 

33 

28 

58023 

81445 

59436 

80420 

60830 

79371 

62206 

78297 

63563 

77199 

32 

29 

58047 

81428 

59459 

80403   60853 

79353 

62229 

78279 

63585 

77181 

31 

30 

58070 

81412 

59482 

80386   60876 

i 

79335 

62251 

78261 

63608 

77162 

30 

31 

58094 

81395 

59506 

80368   60899 

79318 

62274 

78243 

63630 

77144 

29 

32 

58118 

81378 

59529 

80351   60922 

79300 

62297 

78225 

63653 

77125 

28 

33 

58141 

81361 

59552 

80334   60945 

79282 

62320 

78206 

63675 

77107 

27 

34 

58165 

81344 

59576 

80316  ; 

60968 

79264 

62342 

78188 

63698 

77088 

26 

35 

58189 

81327 

59599 

80299  ! 

60991 

79247 

62365 

78170 

63720 

77070 

25 

36 

58212 

81310 

59622 

80282  j  61015 

79229 

62388 

78152 

63742 

77051 

24 

37 

58236 

81293 

59646 

80264 

61038 

79211 

62411 

78134 

63765 

77033 

2a 

38 

58260 

81276 

59669 

80247 

61061 

79193 

€2433 

78116 

63787 

77014 

22 

39 

58283 

81259 

59693 

80230   61084 

79176 

62456 

78098 

63810 

76996 

21 

40 

58307 

81242 

59716 

80212   61107 

79158 

62479 

78079 

63832 

76977 

20 

41 

58330 

81225 

59739 

80195   61130 

79140 

62502 

78061  i 

63854 

76959 

19 

42 

58354 

81208 

59763 

80178   61153 

79122 

62524 

78043 

63877 

76940 

18 

43 

58378 

81191 

59786 

80160   61176 

79105 

62547 

78025  | 

63899 

76921 

17 

44 

58401 

81174 

59809 

80143   61199 

79087 

62570 

78007 

63922 

76903 

16 

45 

58425 

81157 

59832 

80125 

61222 

79069 

62592 

77988 

63944 

76884 

15 

46 

58449 

81140 

59856 

80108 

61245 

79051 

62615 

77970 

63966 

76866   14 

47 

58472 

81123 

59879 

80091 

61268 

79033  ; 

62638 

77952 

63989 

76847   la 

48 

58496 

81106 

59902 

80073  !  61291 

79016  i 

62660 

77934 

64011 

76828 

12 

49 

58519 

81089 

59926 

80056  !  61314 

78998  ' 

62683 

77916 

64033 

76810 

11 

50 

58543 

81072 

59949 

80038  '  61337 

78980 

62706 

77897 

64056 

76791   10 

51 

58567 

81055 

59972 

80021  '  61360 

78962  |  62728 

77879 

64078 

76772 

9 

52 

58590 

81038 

59995 

80003 

61383 

78944 

62751 

77861 

64100 

76754 

8 

53 

58614 

81021 

60019 

79986 

61406 

78926 

62774 

77843 

64123 

76735 

7 

54 

58637 

81004 

60042 

79968 

61429 

78908 

62796 

77824 

64145 

76717 

6 

55 

58661 

80987 

60065 

79951 

61451 

78891 

62819 

77806 

64167 

76698 

5 

56 

58684 

80970 

60089 

79934 

61474 

78873 

62842 

77788 

64190 

76679 

4 

57. 

58708 

80953 

60112 

79916 

61497 

78855 

62864 

77769 

64212 

76661 

3 

58 

58731 

80936 

60135 

79899 

61520 

78837 

62887 

77751 

64234 

76642 

2 

59 

58755 

80919 

60158 

79881 

61543 

78819 

62909 

77733 

64256 

76623 

1 

60 

58779 

80902 

60182 

79864 

61566 

78801 

62932 

77715 

64279 

76604 

0 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

/ 

51° 

53o 

53° 

51° 

50° 

/ 

718 


APPENDIX. 


NATURAL,     SINES     AND     COSINES. 


4,0° 

41° 

43° 

43°          4,4=° 

/ 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

/ 

0 

64279 

76604 

65606 

75471 

66913 

74314 

68200 

73135  1!  69466 

71934 

60 

1 

64301 

76586 

65628 

75452 

66935 

74295   68221 

73116   69487 

71914 

59 

2 

64323 

76567 

65650 

75433 

66956 

74276  j  68242 

73096 

69508 

71894 

58 

3 

64346 

76548 

65672 

75414 

66978 

74256   68264 

73076 

69529 

71873 

57 

4 

64368 

76530 

;  65694 

75395 

66999 

74237 

68285 

73056 

69549 

71853 

56 

5 

64390 

76511 

65716 

75375 

67021 

74217 

68306 

73036 

69570 

71833 

55 

6 

64412 

76492 

65738 

75356 

67043 

74198  |  68327 

73016 

69591 

71813 

54 

7 

64435 

76473 

65759 

75337 

67064 

74178 

68349 

72996 

69612 

71792 

53 

8 

64457 

76455 

65781 

75318 

67086 

74159 

68370 

72976 

69633 

71772 

52 

9 

64479 

76436 

65803 

75299 

67107 

74139 

68391 

72957 

69654 

71752 

51 

10 

64501 

76417 

65825 

75280 

67129 

74120 

68412 

72937 

69675 

71732 

50 

11 

64524 

76398 

65847 

75261 

67151 

74100 

68434 

72917 

69696 

71711 

49 

12 

64546 

76380 

65869 

75241 

67172 

74080 

68455 

72897 

69717 

71691 

48 

13 

64568 

76361 

65891 

75222 

67194 

74061 

68476 

72877 

69737 

71671 

47 

14 

64590 

76342 

65913 

75203 

67215 

74041 

68497 

72857 

69758 

71650 

46 

15 

64612 

76323 

65935 

75184 

67237 

74022 

68518 

72837 

69779 

71630 

45 

16 

64635 

76304 

65956 

75165 

67258 

74002 

68539 

72817 

69800 

71610 

44 

17 

64657 

76286 

65978 

75146 

67280 

73983 

68561 

72797 

69821 

71590 

43 

18 

64679 

76267 

66000 

75126 

67301 

73963 

68582 

72777  !  69842 

71569 

42 

IP 

64701 

76248 

66022 

75107 

67323 

73944 

68603 

72757  '  69862 

71549 

41 

20 

64723 

76229 

66044 

75088 

67344 

73924 

68624 

72737  ::  69883 

71529 

40 

21 

64746 

76210 

66066 

75069 

67366 

73904 

68645 

72717  ]  69904 

71508 

39 

22 

64768 

76192 

66088 

75050 

67387 

73885 

68666 

72697   69925 

71488 

38 

23 

64790 

76173 

66109 

75030 

67409 

73865 

68688 

72677   69946 

71468 

37 

24 

64812 

76154 

66131 

75011 

67430 

73846 

68709 

72657 

69966 

71447 

36 

25 

64834 

76135 

66153 

74992 

67452 

73826 

68730 

72637 

69987 

71427 

35 

26 

64856 

76116 

66175 

74973 

67473 

73806 

68751 

72617 

70008 

71407 

34 

27 

64878 

76097 

66197 

74953 

67495 

73787 

68772 

72597 

70029 

71386 

33 

28 

64901 

76078 

66218 

74934 

67516 

73767 

68793 

72577 

70049 

71366 

32 

29 

64923 

76059 

66240 

74915 

67538 

73747 

68814 

72557 

70070 

71345 

31 

30 

64945 

76041 

66262 

74896 

67559 

73728 

68835 

72537 

70091 

71325 

30 

31 

64967 

76022 

66284 

74876 

67580 

73708 

68857 

72517 

70112 

71305 

29 

32 

64989 

76003 

66306 

74857  ji  67602 

73688 

68878 

72497  ||  70132 

71284 

28 

33 

65011 

75984 

66327 

74838 

67623 

73669 

68899 

72477  I  70153 

71264 

27 

34 

65033 

75965 

66349 

74818 

67645 

73649  ; 

68920 

72457 

70174 

71243 

26 

35 

65055 

75946 

66371 

74799 

67666 

73629 

68941 

72437 

70195 

71223 

25 

36 

65077 

75927 

66393 

74780 

67688 

73610 

68962 

72417 

70215 

71203 

24 

37 

65100 

75908 

66414 

74760 

67709 

73590 

68983 

72397 

70236 

71182 

23 

38 

65122 

75889 

66436 

74741* 

67730 

73570 

69004 

72377 

-70257 

71162 

22 

39 

65144 

75870 

66458 

74722 

67752 

73551 

69025 

72357 

70277 

71141 

21 

40 

65166 

75851 

66480 

74703 

67773 

73531 

69046 

72337 

70298 

71121 

20 

41 

65188 

75832 

66501 

74683 

67795 

73511 

69067 

72317 

!  70319 

71100 

19 

42 

65210 

75813 

66523 

74664 

67816 

73491 

69088 

72297 

i  70339 

71080 

18 

43 

65232 

75794 

66545 

74644 

67837 

73472 

69109 

72277 

!  70360 

71059 

17 

44 

65254 

75775 

66566 

74625 

67859 

73452 

69130 

72257 

!  70381 

71039 

16 

45 

65276 

75756 

66588 

74606 

67880 

73432 

69151 

72236 

70401 

71019 

15 

46 

65298 

75738 

66610 

74586 

67901 

73413 

69172 

72216 

70422 

70998 

14 

47 

65320 

75719 

66632 

74567 

67923 

73393 

69193 

72196 

70443 

70978 

13 

48 

65342 

75700 

66653 

74548 

67944 

73373 

69214 

72176 

70463 

70957 

12 

49 

65364 

75680 

66675 

74528 

67965 

73353 

69235 

72156 

70484 

70937 

11 

50 

65386 

75661 

66697 

74509 

67987 

73333 

69256 

72136  :!  70505 

70916 

10 

51 

65408 

75642 

66718 

74489 

68008 

73314 

69277 

72116 

70525 

70896 

9 

52 

65430 

75623 

66740 

74470 

68029 

73294 

69298 

72095 

70546 

70875 

8 

53 

65452 

75604 

66762 

74451 

68051 

73274 

69319 

72075 

70567 

70855 

7 

54 

65474 

75585 

66783 

74431 

68072 

73254 

69340 

72055 

70587 

70834 

6 

55 

65496 

75566 

66805 

74412 

68093 

73234 

69361 

72035 

70608 

70813 

5 

56 

65518 

75547 

66827 

74392 

68115 

73215 

69382 

72015 

70628 

70793 

4 

57 

65540 

75528 

66848 

74373 

68136 

73195 

69403 

71995 

70649 

70772 

3 

58 

65562 

75509 

66870 

74353 

68157 

73175 

69424 

71974 

70670 

70752 

2 

59 

65584 

75490 

66891 

74334 

68179 

73155 

69445 

71954 

70690 

70731 

1 

60 

65606 

75471 

66913 

74314 

68200 

73135  i 

69466 

71934 

70711 

70711 

0 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

/ 

4O° 

48° 

47° 

40° 

45° 

/ 

APPENI)IX.//S 

. 

LOGARITHMS     OP     NUMBERS. 


719 


IV. 

o 

1 

3 

3 

-A 

5 

0 

7 

S 

9 

I>. 

100 

00  0000 

0434 

0868 

1301 

1734 

2166 

2598 

3029 

3461 

3891 

432 

101 

4321 

4751 

5181 

5609 

6038 

6466 

6894 

7321 

7748 

8174 

428 

102 

*  8600 

9026 

9451 

9876 

+300 

0724 

1147 

1570 

1993 

2415 

424 

103 

01  2837 

3259 

3680 

4100 

4521 

4940 

5360 

5779 

6197 

6616 

419 

104 

*  7033 

7451 

7868 

8284 

8700 

9116 

9532 

9947 

+361 

0775 

416 

105 

02  1189 

1603 

2016 

2428 

2841 

3252 

3664 

4075 

4486 

4896 

412 

106 

5306 

5715 

6125 

6533 

6942 

7350 

7757 

8164 

8571 

8978 

408 

107 

*  9384 

9789 

4195 

0600 

1004 

1408 

1812 

2216 

2619 

3021 

404 

108 

03  3424 

3826 

4227 

4628 

5029 

5430 

5830 

6230 

6629 

7028 

400 

109 

*  7426 

7825 

8223 

8620 

9017 

9414 

9811 

+207 

0602 

0998 

396 

110 

04  1393 

1787 

2182 

2576 

2969 

3362 

3755 

4148 

4540 

4932 

393 

111 

5323 

5714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

389 

112 

*  9218 

9606 

9993 

*380 

0766 

1153 

1538 

1924 

2309 

2694 

386 

113 

05  3078 

3463 

3846 

4230 

4613 

4996 

5378 

5760 

6142 

6524 

382 

114 

*  6905 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

+320 

379 

115 

06  0698 

1075 

1452 

1829 

2206 

2582 

2958 

3333 

3709 

4083 

376 

116 

4458 

4832 

5206 

5580 

5953 

6326 

6699 

7071 

7443 

7815 

372 

117 

*  8186 

8557 

8928 

9298 

9668 

+038 

0407 

0776 

1145 

1514 

369 

118 

07  1882 

2250 

2617 

2985 

3352 

3718 

4085 

4451 

4816 

5182 

366 

119 

5547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

363 

120 

*  9181 

9543 

9904 

+266 

0626 

0987 

1347 

1707 

2067 

2426 

360 

121 

08  2785 

3144 

3503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

357 

122 

6360 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

355 

123 

*  9905 

+258 

0611 

0963 

1315 

1667 

2018 

2370 

2721 

3071 

351 

124 

09  3422 

3772 

4122 

4471 

4820  , 

5169 

5518 

5866 

6215 

6562 

349 

125 

*  6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

+026 

346 

126 

10  0371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

343 

127 

3804 

4146 

4487 

4828 

5169 

5510 

5851 

6191 

6531 

6871 

340 

128 

*  7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

+253 

338 

129 

11  0590 

0926 

1263 

1599 

1934 

2270 

2605 

2940 

3275 

3609 

335 

130 

3943 

4277 

4611 

4944 

5278 

5611 

5943 

6276 

6608 

6940 

333 

131 

*  7271 

7603 

7934 

8265 

8595 

8926 

9256 

9586 

9915 

+245 

330 

132 

12  0574 

0903 

1231 

1560 

1888 

2216 

2544 

2871 

3198 

3525 

328 

133 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

6131 

6456 

6781 

325 

134 

*  7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

+012 

323 

135 

13  0334 

0655 

0977 

1298 

1619 

1939 

2260 

2580 

2900 

3219 

•321 

136 

3539 

3858 

4177 

4496 

4814 

5133 

5451 

5769 

6086 

6403 

318 

137 

6721 

7037 

7354 

7671 

7987 

8303 

8618 

8934 

9249 

9564 

315 

138 

*9879 

4194 

0508 

0822 

1136 

1450 

1763 

2076 

2389 

2702 

314 

139 

143015  ' 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507 

5818 

311 

140 

6128 

6438 

6748 

7058 

7367 

7676 

7985 

8294 

8603 

8911 

309 

141 

*9219 

9527 

9835 

+142 

0449 

0756 

1063 

1370 

1676 

1982 

307 

142 

15  2288 

2594 

2900 

3205 

3510 

3815 

4120 

4424 

4728 

5032 

305 

143 

5336 

5640 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

303 

144 

*  8362 

8664 

8965 

9266 

9567 

9868 

+168 

0469 

0769 

1068 

301 

145 

16  1368 

1667 

1967 

2266 

2564 

2863 

3161 

3460 

3758 

4055 

299 

146 

4353 

4650 

4947 

5244 

5541 

5838 

6134 

6430 

6726 

7022 

297 

147 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

295 

148 

17  0262 

0555 

0848 

1141 

1434 

1726 

2019 

2311 

2603 

2895 

293 

149 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

291 

150 

6091 

6381 

6670 

6959 

7248 

7536 

-7825 

8113 

8401 

8689 

289 

151 

*  8977 

9264 

9552 

9839 

+126 

0413 

0699 

0985 

1272 

1558 

287 

152 

18  1844 

2129 

2415 

2700 

2985 

3270 

3555 

3839 

4123 

4407 

285 

153 

4691 

4975 

5259 

5542 

5825 

6108 

6391 

6674 

6956 

7239 

283 

154 

*7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

+051 

281 

155 

19  0332 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2567 

2846 

279 

156 

3125 

3403 

3681 

3959 

4237 

4514 

4792 

5069 

5346 

5623 

278 

157 

5900 

6176 

6453 

6729 

7005 

7281 

7556 

7832 

8107 

8382 

276 

158 

*  8657 

8932 

9206 

9481 

9755 

+029 

0303 

0577 

0850 

1124 

274 

159 

20  1397 

1670 

1943 

2216 

2488 

2761 

3033 

3305 

8577 

3848 

272 

IV. 

O 

1 

2 

3 

4, 

5 

G 

7 

8 

9 

j>. 

720 


APPENDIX. 

LOGARITHMS    OP     NUMBERS. 


2V. 

0 

1 

3 

3 

4, 

5 

0 

7 

8 

O 

r>. 

160 

20  4120 

4391 

4663 

4934 

5204 

5475 

5746 

6016 

6286 

6556 

271 

161 

6826 

7096 

7365 

7634 

7904 

8173 

8441 

8710 

8979 

9247 

269 

162 

*  9515 

9783 

4051 

0319 

0586 

0853 

1121 

1388 

1654 

1921 

267 

163 

21  2188 

2454 

2720 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

266 

164 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264 

165 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

262 

166 

22  0108 

0370 

0631 

0892 

1153 

1414 

1675 

1936 

2196 

2456 

261 

167 

2716 

2976 

3236 

3496 

3755 

4015 

4274 

4533 

4792 

5051 

259 

168 

5309 

5568 

5826 

6084 

6342 

6600 

6858 

7115 

7372 

7630 

258 

169 

*7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

4193 

256 

170 

23  0449 

0704 

0960 

1215 

1470 

1724 

1979 

2234 

2488 

2742 

254 

171 

2996 

3250 

3504 

3757 

4011 

4264 

4517 

4770 

5023 

5276 

253 

172 

5528 

5781 

6033 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

252 

173 

*8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

4050 

0300 

250 

174 

24  0549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541 

2790 

249 

175 

3038 

3286 

3534 

3782 

4030 

4277 

4525 

4772 

5019 

5266 

248 

176 

5513 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

177 

*7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

+176 

245 

178 

25  0420 

0664 

0908 

1151 

1395 

1638 

1881 

2125 

2368 

2610 

243 

179 

2853 

3096 

3338 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

242 

180 

5273 

5514 

5755 

5996 

6237 

6477 

6718 

6958 

7198 

7439 

241 

181 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

182 

26  0071 

0310 

0548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

238 

183 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

184 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

185 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

186 

*  9513 

9746 

9980 

+213 

0446 

0679 

0912 

1144 

1377 

1609 

233 

187 

27  1842 

2074 

2306 

2538 

2770 

3001 

3233 

3464 

3696 

3927 

232 

188 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

230 

189 

6462 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

8525 

22$ 

190 

*  8754 

8982 

9211 

9439 

9667 

9895 

*123 

0351 

0578 

0806 

228 

191 

28  1033 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2849 

3075 

227 

192 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882 

5107 

5332 

22$ 

193 

5557 

5782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7578 

225 

194 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

9589 

9812 

223 

195 

29  0035 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

1813 

2034 

222 

196 

2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

4025 

4246 

221 

197 

4466 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

198 

6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

199 

*  8853 

9071 

9289 

9507 

9725 

9943 

4161 

0378 

0595 

0813 

218 

200 

30  1030 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

217 

201 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

216- 

202 

5351 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

215 

203 

7496 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

2ia 

204 

*  9630 

9843 

4056 

0268 

0481 

0693 

0906 

1118 

1330 

1542 

212 

205 

31  1754 

1966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

211 

206 

3867 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

210 

207 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

209 

208 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

208 

209 

32  0146 

0354 

0562 

0769 

0977 

1184 

1391 

1598 

1805 

2012 

207 

210 

2219 

2426 

2633 

2839 

3046 

3252 

3458 

3665 

3871 

4077 

206 

211 

4282 

4488 

4694 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

205 

212 

6336 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204 

213 

*8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

4008 

0211 

203 

214 

33  0414 

0617 

0819 

1022 

1225 

1427 

1630 

1832 

2034 

2236 

202 

215 

2438 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

4253 

202 

216 

4454 

4655 

4856 

5057 

5257 

5458 

5658 

5859 

6059 

6260 

201 

217 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

200 

218 

*  8456 

8656 

8855 

9054 

9253 

9451 

9650 

9849 

4047 

0246 

199 

219 

34  0444 

0642 

0841 

1039 

1237 

1435 

1632 

1830 

2028 

2225 

198 

3V. 

0 

1 

3 

3 

4      5 

078 

0 

r>* 

APPENDIX. 


721 


LOGARITHMS    OP     NUMBERS. 


N. 

013 

3 

4, 

5 

O      7 

8 

9 

r>. 

220 

34  2423    2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

197 

221 

4392  i  4589 

4785 

4981 

5178 

5374 

5570 

5766 

5962 

6157 

196 

222 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

223 

*  8305 

8500   8694 

8889 

9083 

9278 

9472 

9666 

9860 

+054 

194 

224 

35  0248 

0442 

0636 

0829 

1023 

1216 

1410 

1603 

1796 

1989 

193 

225 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

193 

226 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

192 

227 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7554 

7744 

191 

228 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9456 

9646 

190 

229 

*  9835 

+025 

0215 

0404 

0593 

0783 

0972 

1161 

1350 

1539 

189 

230 

36  1728 

1917 

2105 

2294 

2482 

2671 

2859 

3048 

3236 

3424 

188 

231 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113 

5301 

188 

232 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

233 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

186 

234 

*  9216 

9401 

9587 

9772 

9958 

+143 

0328 

0513 

0698 

0883 

185 

235 

37  1068 

1253 

1437 

1622 

1806 

1991 

2175 

2360 

2544 

2728 

184 

236 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

237 

4748  !  4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

183 

238 

6577  i  6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182 

239 

*  8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

+030 

181 

240 

38  0211 

0392 

0573 

0754 

0934 

1115 

1296 

1476 

1656 

1837 

181 

241 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

180 

242 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

5249 

5428 

179 

243 

5606 

5785 

5964  !  6142 

6321 

6499 

6677 

6856 

7034 

7212 

178 

244 

7390 

7568 

7746 

7923 

8101 

8279 

8456 

8634 

8811 

8989 

178 

245 

*  9166 

9343 

9520 

9698 

9875 

+051 

0228 

0405 

0582 

0759 

177 

246 

39  0935 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345 

2521 

176 

247 

2697 

2873 

3048 

3224 

3400 

3575 

3751 

3926 

4101 

4277 

176 

248 

4452 

4627 

4802 

4977 

5152 

5326 

5501 

5676 

5850 

6025 

175 

249 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

250 

7940 

8114 

8287 

8461 

8634 

8808 

8981 

9154 

9328 

9501 

173 

251 

*  9674 

9847 

+020 

0192 

0365 

0538 

0711 

0883 

1056 

1228 

173 

252 

40  1401 

1573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

172 

253 

3121 

3292 

3464 

3635 

3807 

3978 

4149 

4320 

4492 

4663 

171 

254 

4834 

5005 

5176 

5346 

5517 

5688 

5858 

6029 

6199 

6370 

171 

255 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

170 

256 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

169 

257 

*  9933 

+102 

0271 

0440 

0609 

0777 

0946 

1114 

1283 

1451 

169 

258 

41  1620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

168 

259 

3300 

3467 

3635 

3803 

3970 

4137 

4305 

4472 

4639 

4806 

167 

260 

4973 

5140 

5307 

5474 

5641 

5808 

5974 

6141 

6308 

6474 

167 

261 

6641 

6807 

6973 

7139 

7306 

7472 

7638 

7804 

7970 

8135 

166 

262 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

165 

263 

*  9956 

+121 

0286 

0451 

0616 

0781 

0945 

1110 

1275 

1439 

165 

264 

42  1604 

1768 

1933 

2097 

2261 

2426 

2590 

2754 

2918 

3082 

164 

265 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

266 

4882 

5045 

5208 

5371 

5534 

5697 

5860 

6023 

6186 

6349 

iea 

267 

6511 

6674 

6836 

6999 

7161 

7324 

7486 

7648 

7811 

7973 

162, 

268 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

162 

269 

*9752 

9914 

+075 

0236 

0398 

0559 

0720 

0881 

1042 

1203 

161 

270 

43  1364 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

161 

271 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

4249 

4409 

160' 

272 

4569 

4729 

4888 

5048 

5207 

5367 

5526 

5685 

5844 

6004 

159' 

273 

6163 

6322 

6481 

6640 

6799 

6957 

7116 

7275 

7433 

7592 

159' 

274 

7751 

7909 

8067 

8226 

8384 

8542 

8701 

8859 

9017 

9175 

158 

275 

*  9333 

9491 

9648 

9806 

9964 

+122 

0279 

0437 

0594 

0752 

158 

276 

44  0909 

1066 

1224 

1381 

1538 

1695 

1852 

2009 

2166 

2323 

157 

277 

2480 

2637 

2793 

2950 

3106 

3263 

3419 

3576 

3732 

3889 

157 

278 

4045 

4201 

4357 

4513 

4669 

4825 

4981 

5137 

5293 

5449 

156 

279 

5604 

5760 

5915 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

155 

IV. 

0      1 

3 

3 

4, 

5 

O 

.  7 

H 

0 

r>* 

722 


APPENDIX. 


LOGARITHMS     OP     NUMBERS. 


IV. 

0 

1 

3 

3 

4, 

5 

G 

7 

8 

9 

I>. 

280 

44  7158 

7313 

7468 

7623 

7778   7933 

8088 

8242 

8397 

8552 

155 

281 

*  8706 

8861 

9015 

9170 

9324 

9478 

9633 

9787 

9941 

+095 

154 

282 

45  0249 

0403 

0557 

0711 

0865 

1018 

1172 

1326 

1479 

1633 

154 

283 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

153 

284 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

285 

4845 

4997 

5150 

5302 

5454 

5606 

5758 

5910 

6062 

6214 

152 

286 

6366 

6518 

6670 

6821 

6973 

7125 

7276 

7428 

7579 

7731 

152 

287 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

151 

288 

*.9392 

9543 

9694 

9845 

9995 

+146 

0296 

0447 

0597 

0748 

151 

289 

46  0898 

1048 

1198 

1348 

1499 

1649 

1799 

1948 

2098 

2248 

150 

290 

2398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

150 

291 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

5085 

5234 

149 

292 

5383 

5532 

5680 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

293 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

148 

294 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9527 

9675 

148 

295 

*  9822 

9969 

*116 

0263 

0410 

0557 

0704 

0851 

0998 

1145 

147 

296 

47  1292 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

146 

297 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

298 

4216 

4362 

4508 

4653 

4799 

4944 

5090 

5235 

5381 

5526 

146 

299 

5671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

145 

300 

7121 

7266 

7411 

7555 

7700 

7844 

7989 

8133 

8278 

8422 

145 

301 

8566 

8711 

8855 

'8999 

9143 

9287 

9431 

9575 

9719 

9863 

144 

302 

48  0007 

0151 

0294 

0438 

0582 

0725 

0869 

1012 

1156 

1299 

144 

303 

1443 

1586 

1729 

1872 

2016 

2159 

2302 

2445 

2588 

2731 

143 

304 

2874 

3016 

3159 

3302 

3445 

3587 

3730 

3872 

4015 

4157 

143 

305 

4300 

4442 

4585 

4727 

4869 

5011 

5153 

5295 

5437 

5579 

142 

306 

5721 

5863 

6005 

6147 

6289 

6430 

6572 

6714 

6855 

6997 

142 

307 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

8410 

141 

308 

8551 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9677 

9818 

141 

309 

*  9958 

+099 

0239 

0380 

0520 

0661 

0801 

0941 

1081 

1222 

140 

310 

49  1362 

1502 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2621 

140 

311 

2760 

2900 

3040 

3179 

3319 

3458 

3597 

3737 

3876 

4015 

139 

312 

4155 

4294 

4433 

4572 

4711 

4850 

4989 

5128 

5267 

5406 

139 

313 

5544 

5683 

5822 

5960 

6099 

6238 

6376 

6515 

6653 

6791 

139 

314 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

138 

315 

8311 

8448 

8586 

8724 

8862 

8999 

9137 

9275 

9412 

9550 

138 

316 

*  9687 

9824 

9962 

+099 

0236 

0374 

0511 

0648 

0785 

0922 

137 

317 

50  1059 

1196 

1333 

1470 

1607 

1744 

1880 

2017 

2154 

2291 

137 

318 

2427 

2564 

2700 

2837 

2973 

3109 

3246 

3382 

3518 

3655 

136 

319 

3791 

3927 

4063 

4199 

4335 

4471 

4607 

4743 

4878 

5014 

136 

320 

5150 

5286 

5421 

5557 

5693 

5828 

5964 

6099 

6234 

6370 

136 

321 

6505 

6640 

6776 

6911 

7046 

7181 

7316 

7451 

7586 

7721 

135 

322 

7856 

7991 

8126 

8260 

8395 

8530 

8664 

8799 

8934 

9068 

135 

323 

*  9203 

9337 

9471 

9606 

9740 

9874 

+009 

0143 

0277 

0411 

134 

324 

51  0545 

0679 

0813 

0947 

1081 

1215 

1349 

1482 

1616 

1750 

134 

325 

1883 

2017 

2151 

2284 

2418 

2551 

2684 

2818 

2951 

3084 

133 

326 

3218 

3351 

3484 

3617 

3750 

3883 

4016 

4149 

4282 

4414 

133 

327 

4548 

4681 

4813 

4946 

5079 

5211 

5344 

5476 

5609 

5741 

133 

328 

5874 

6006 

6139 

6271 

6403 

6535 

6668 

6800 

6932 

7064 

132 

329 

7196 

7328 

7460 

7592 

7724 

7855 

7987 

8119 

8251 

8382 

132 

330 

8514 

8646 

8777 

8909 

9040 

9171 

9303 

9434 

9566 

9697 

131 

331 

*9828 

9959 

+090 

0221 

0353 

0484 

0615 

0745 

0876 

1007 

131 

332 

52  1138 

1269 

1400 

1530 

1661 

1792 

1922 

2053 

2183 

2314 

131 

333 

2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

3616 

130 

334 

3746 

3876 

4006 

4136 

4266 

4396 

4526 

4656 

4785 

4915 

130 

335 

5045 

5174 

5304 

5434 

5563 

5693 

5822 

5951 

6081 

6210 

129 

336 

6339 

6469 

6598 

6727 

6856 

6985 

7114 

7243 

7372 

7501 

129 

337 

7630 

7759 

7888 

8016 

8145 

8274 

8402 

8531 

8660 

8788 

129 

338 

*  8917 

9045 

9174 

9302 

9430 

9559 

9687 

9815 

9943 

+072 

128 

339 

53  0200 

0328 

0456 

0584 

0712 

0840 

0968 

1096 

1223 

1351 

128 

IV. 

0 

1 

3 

3 

4= 

5 

O 

7 

8 

9 

r>. 

APPENDIX. 


723 


LOGARITHMS     OF     NUMBERS. 


IV. 

o 

1 

3 

3 

4, 

5 

6      7 

8 

9 

I>. 

340 

53  1479 

1607 

1734 

1862 

1990 

2117 

2245 

2372 

2500 

2627 

128 

541 

2754 

2882 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3899 

127 

342 

4026 

4153 

4280 

4407 

4534 

4661 

4787 

4914 

5041 

5167 

127 

343 

5294 

5421 

5547 

5674 

5800 

5927 

6053 

6180 

6306 

6432 

126 

344 

6558 

6685 

6811 

6937 

7063 

7189 

7315 

7441 

7567 

7693 

126 

345 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

126 

346 

*  9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

+079 

0204 

125 

347 

54  0329 

0455 

0580 

0705 

0830 

0955 

1080 

1205 

1330 

1454 

125 

348 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

349 

2825 

2950 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

124 

350 

4068 

4192 

4316 

4440 

4564 

4688 

4812 

4936 

5060 

5183 

124 

351 

5307 

5431 

5555 

5678 

5802 

5925 

6049 

6172 

6296 

6419 

124 

352 

6543 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

123 

353 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

123 

354 

*9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

+106 

123 

355 

55  0228 

0351 

0473 

0595 

0717 

0840 

0962 

1084 

1206 

1328 

122 

356 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

122 

357 

2668 

2790 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

121 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

121 

359 

5094 

5215 

5336 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

121 

360 

6303 

6423 

6544 

6664 

6785 

6905 

7026 

7146 

7267 

7387 

120 

361 

7507 

7627 

7748  |  7868 

7988 

8108 

8228 

8349 

8469 

8589 

120 

362 

8709 

8829 

8948    9068 

9188 

9308 

9428 

9548 

9667 

9787 

120 

363 

*  9907 

+026 

0146   0265 

0385 

0504 

0624 

0743 

0863 

0982 

119 

364 

56  1101 

1221 

1340   1459 

1578 

1698 

1817 

1936 

2055 

2174 

119 

365 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

119 

366 

3481 

3600 

3718 

3837 

3955 

4074 

4192 

4311 

4429 

4548 

119 

367 

4666 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

5730 

118 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

118 

369 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

370 

8202 

8319 

8436 

8554 

8671 

8788 

8905 

9023 

9140 

9257 

117 

371 

*  9374 

9491 

9608 

9725 

9842 

9959 

+076 

0193 

0309 

0426 

117 

372 

57  0543 

0660 

0776 

0893 

1010 

1126 

1243 

1359 

1476 

1592 

117 

373 

1709 

1825    1942 

2058 

2174 

2291 

2407 

2523 

2639 

2755 

116 

374 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

3684 

3800 

3915 

116 

375 

4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116 

376 

5188 

5303 

5419 

5534 

5650 

5765 

5880 

5996 

6111 

6226 

115 

377 

6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

115 

378 

7492 

7607 

7722 

7836 

7951 

8066 

8181 

8295 

8410 

8525 

115 

379 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

114 

380 

*9784 

9898 

+012 

0126 

0241 

0355 

0469 

0583 

0697 

0811 

114 

381 

58  0925 

1039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1950 

114 

382 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

2972 

3085 

114 

383 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4105 

4218 

113 

384 

4331 

4444 

4557 

4670 

4783 

4896 

5009 

5122 

5235 

5348 

113 

385 

5461 

5574 

5686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

113 

386 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

112 

387 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

112 

388 

8832 

8944 

9056 

9167 

9279 

9391 

9503 

9615 

9726 

9838 

112 

389 

*  9950 

+061 

0173 

0284 

0396 

0507 

0619 

0730 

0842 

0953 

112 

390 

59  1065 

1176 

1287 

1399 

1510 

1621 

1732 

1843 

1955 

2066 

111 

391 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3175 

111 

392 

3286 

3397 

3508 

3618 

3729 

3840 

3950 

4061 

4171 

4282 

111 

393 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

5386 

110 

394 

5496 

5606 

5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

110 

395 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

110 

396 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

110 

397 

8791 

8900 

9009 

9119 

9228 

9337 

9446 

9556 

9665 

9774 

109 

398 

*  9883 

9992 

+101 

0210 

0319 

0428 

0537 

0646 

0755 

0864 

109 

399 

60  0973 

1082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

1951 

109 

N. 

O 

1 

2 

3 

4 

5 

G     7 

S 

0 

r>. 

724 


APPENDIX. 


LOGARITHMS    OF     NUMBERS. 


IV. 

0 

1 

2 

3 

4, 

5 

0 

7 

8 

9 

r>. 

400 

60  2060 

2169 

2277   2386 

2494 

2603 

2711 

2819 

2928 

3036 

108- 

401 

3144 

3253 

3361 

3469 

3577 

3686 

3794 

3902 

4010 

4118 

108 

402     4226 

4334 

4442 

4550 

4658 

4766 

4874 

4982 

5089 

5197 

108 

403 

5305 

5413 

5521 

5628 

5736 

5844 

5951 

6059 

6166 

6274 

108 

404 

6381 

6489 

6596 

6704 

6811 

6919 

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3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

58 

748 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

58 

749 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

5003 

58 

750 

5061 

5119 

5177 

5235 

5293 

5351 

5409 

5466 

5524 

5582 

58 

751 

5640 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

58 

752 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

58 

753 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

58 

754 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

58 

755 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

57 

756 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

57 

757 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

57 

758 

*  9669 

9726 

9784 

9841 

9898 

9956 

+013 

0070 

0127 

0185 

57 

759 

88  0242 

0299 

0356 

0413 

0471 

0528 

0585 

0642 

0699 

0756 

57 

IV. 

0 

1 

3 

3 

4= 

5 

6 

7 

S 

e 

r>. 

730 


APPENDIX. 


LOGARITHMS    OF    NUMBERS. 


]V. 

O 

1 

2      3 

4.      5 

G 

7 

8 

O 

r>» 

760 

88  0814 

0871 

0928 

0985 

1042 

1099 

1156 

1213 

1271 

1328 

57 

761 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

57 

762 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

57 

763 

2525 

2581 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

57 

764 

3093 

3150 

3207 

3264 

3321 

3377 

3434 

3491 

3548 

3605 

57 

765 

3661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

57 

766 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4625 

4682 

4739 

57 

767 

4795 

4852 

4909 

4965 

5022 

5078 

5135 

5192 

5248 

5305 

57 

768 

5361 

5418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

57 

769 

5926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

56 

770 

6491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

56 

771 

7054 

7111 

7167 

7223 

7280 

7336 

7392 

7449 

7505 

7561 

56 

772 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

56 

773 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8685 

56 

774 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

56 

775 

9302 

9358 

9414 

9470 

9526 

9582 

9638 

9694 

9750 

9806 

56 

776 

*9862 

9918 

9974 

+030 

0086 

0141 

0197 

0253 

0309 

0365 

56 

777 

89  0421 

0477 

0533 

0589 

0645 

0700 

0756 

0812 

0868 

0924 

56 

778 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

56 

779 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

56 

780 

2095 

2150 

2206 

2262 

2317 

2373 

2429 

2484 

2540 

2595 

56 

781 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

56 

782 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

56 

783 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

55 

784 

4316 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

55 

785 

4870 

4925 

4980 

5036 

5091 

5146 

5201 

5257 

5312 

5367 

55 

786 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

55 

787 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

55 

788 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

55 

789 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

55 

790 

7627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

55 

791 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

55 

792 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

55 

793 

9273 

9328 

9383 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

55 

794 

*  9821 

9875 

9930 

9985 

+039 

0094 

0149 

0203 

0258 

0312 

55 

795 

90  0367 

0422 

0476 

0531 

0586 

0640 

0695 

0749 

0804 

0859 

55 

796 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

55 

797 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

54 

798 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

54 

799 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

54 

800 

3090 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

3524 

3578 

54 

801 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

54 

802 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

54 

803 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

5202 

54 

804 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

54 

805 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

54 

806 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

54 

807 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

54 

808 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

54 

809 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

54 

810 

8485 

8539 

8592 

8646 

8699 

8753 

8807 

8860 

8914 

8967 

54 

811 

9021 

9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449 

9503 

54 

812 

*  9556 

9610 

9663 

9716 

9770 

9823 

9877 

9930 

9984 

+037 

53 

813 

91  0091 

0144 

0197 

0251 

0304 

0358 

0411 

0464 

0518 

0571 

53 

814 

0624 

0678 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

53 

815 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

53 

816 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2116 

2169 

53 

817 

2222 

2275 

2328 

2381 

2435 

2488 

2541 

2594 

2647 

2700 

53 

818 

2753 

2806 

2859 

2913 

2966 

3019 

3072 

3125 

3178 

3231 

53 

819 

3284 

3337 

3390 

3443 

3496 

3549 

3602 

3655 

3708 

3761 

53 

3V. 

O 

1 

3 

3 

4= 

5 

0 

y 

8 

9 

r>. 

APPENDIX. 


731 


LOGARITHMS     OP     NUMBERS. 


NT. 

O 

1 

2 

3 

4, 

5 

6      7 

8 

O 

r>. 

820 

91  3814 

3867 

3920 

3973 

4026 

4079 

4132 

4184 

4237 

4290 

53 

821 

4343 

4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

53 

822 

4872 

4925 

4977 

5030 

5083 

5136 

5189 

5241 

5294 

5347   53 

823 

5400 

5453 

5505 

5558 

5611 

5664 

5716 

5769 

5822 

5875 

53 

824 

5927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

53 

825 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

53 

826 

6980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

53 

827 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

52 

828 

8030 

8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

52 

829 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

52 

830 

9078 

9130 

9183 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

52 

831 

*  9601 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

+019 

0071 

52 

832 

92  0123 

0176 

0228 

0280 

0332 

0384 

0436 

0489 

0541 

0593 

52 

833 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

52 

834 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

52 

835 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

52 

836 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

52 

837 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

52 

838 

3244 

3296 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

52 

839 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

52 

840 

4279 

4331 

4383 

4434 

4486 

4538 

4589 

4641 

4693 

4744 

52 

841 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5261 

52 

842 

5312 

5364   5415 

5467 

5518 

5570 

5621 

5673 

5725 

5776 

52 

843 

5828 

5879 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

51 

844 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

51 

845 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

51 

846 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

51 

847 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345   51 

848 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857   51 

849 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

51 

850 

9419 

9470 

9521 

9572 

9623 

9674 

9725 

9776 

9827 

9879 

51 

851 

*  9930 

9981 

+032 

0083 

0134 

0185 

0236 

0287 

0338 

0389 

51 

852 

93  0440 

0491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

51 

853 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

51 

854 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

51 

855 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

51 

856 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

51 

857 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

51 

858 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

51 

859 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

51 

860 

4498 

4549 

4599 

4650 

4700 

4751 

4801 

4852 

4902 

4953 

50 

861 

5003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

50 

862 

5507 

5558 

5608 

5658 

5709 

5759 

5809 

5860 

5910 

5960 

50 

863 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

50 

864 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

50 

865 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

50 

866 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

50 

867 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

50 

868 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970  !  50 

869 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

50 

870 

9519 

9569 

9619 

9669 

9719 

9769 

9819 

9869 

9918 

9965 

50 

871 

94  0018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

0417 

0467 

50 

872 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

50 

873 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

50 

874 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1958 

50 

875 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

50 

876 

2504 

2554 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

2950 

50 

877 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

49 

878 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

49 

879 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

49 

W. 

O 

1 

9 

3 

4= 

5 

6 

7 

8 

0 

r>. 

732 


APPENDIX. 


LOGARITHMS    OP     NUMBERS. 


IV. 

O 

1 

2 

3 

4= 

5 

0 

7 

8     9 

I>. 

880 

94  4483 

4532 

4581 

4631 

4680 

4729 

4779 

4828 

4877  I  4927 

49 

881 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

49 

882 

5469 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

5862 

5912 

49 

883 

5961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

49 

884 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

49 

885 

6943 

6992 

7041 

7090 

7140 

7189 

7238 

7287 

7336 

7385 

49 

886 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

49 

887 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

49 

888 

8413 

8462 

8511 

8560 

8609 

8657 

8706 

8755 

8804 

8853 

49 

889 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

49 

890 

9390 

9439 

9488 

9536 

9585 

9634 

9683 

9731 

9780 

9829 

49 

891 

*  9878 

9926 

9975 

+024 

0073 

0121 

0170 

0219 

0267 

0316 

49 

892 

95  0365 

0414 

0462 

0511 

0560 

0608 

0657 

0706 

0754 

0803 

49 

893 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

49 

894 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

49 

895 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

48 

896 

2308 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

2696 

2744 

48 

897 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

48 

898 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

48 

899 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

48 

ii 

900 

4243 

4291 

4339 

4387 

4435 

4484 

4532 

4580 

4628 

4677 

48 

901 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

5110 

5158 

48 

902 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

48 

903 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

48 

904 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6553 

6601 

48 

905 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48 

906 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7559 

48 

907 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

48 

908 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

48 

909 

8564 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

48 

910 

9041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

9471 

48 

911 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

48 

912 

*  9995 

+042 

0090 

0138 

0185 

0233 

0280 

0328 

0376 

0423 

48 

913 

96  0471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

48 

914 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

47 

915 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

47 

916 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

47 

917 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

47 

918 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

47 

919 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

47 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

47 

921 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

47 

922 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

5108 

5155 

47 

923 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5625 

47 

924 

5672 

5719 

5766 

5813 

5860 

5907 

5954 

6001 

6048 

6095 

47 

925 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

47 

926 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

47 

927 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

47 

928 

7548 

7595 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

7969 

47 

929 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

47 

930 

8483 

8530 

8576 

8623 

8670 

8716 

8763 

8810 

8856 

8903 

47 

931 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

47 

932 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

47 

933 

*  9882 

9928 

9975 

+021 

0068 

0114 

0161 

0207 

0254 

0300 

47 

934 

97  0347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

46 

935 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

46 

936 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

46 

937 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2064 

2110 

2157 

46 

938 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

46 

939 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

46 

N. 

O 

1 

2 

3 

4 

.5 

O 

7 

S      0 

r>. 

APPENDIX. 


733 


LOGARITHMS     OF     NUMBERS. 


N. 

0 

1 

2 

3 

4: 

5 

O 

7 

8 

o  !  r>. 

940 

97  3128 

3174 

3220 

3266 

3313 

3359 

3405 

3451 

3497 

3543 

46 

941 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

46 

942 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

46 

943 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

46 

944 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

46 

945 

5432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

46 

946 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

46 

947 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

46 

948 

6808  i  6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

46 

949 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

46 

950 

7724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

46 

951 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

46 

952 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

46 

953 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

46 

954 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

46 

955 

98  0003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

45 

956 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

45 

957 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

45 

958 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

45 

959 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

45 

960 

2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

45 

961 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

45 

962 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

45 

963 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

45 

964 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

45 

965 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

966 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

45 

967 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

5741 

5786 

5830 

45 

968 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

45 

969 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

45 

970 

6772 

6817 

6861 

6906 

6951 

6996 

7040 

7085 

7130 

7175 

45 

971 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

45 

972 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

45 

973 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

45 

974 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

45 

975 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

45 

976 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

44 

977 

*  9895 

9939 

9983 

+028 

0072 

0117 

0161 

0206 

0250 

0294 

44 

978 

99  0339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

44 

979 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

44 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

44 

981 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

44 

982 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

44 

983 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

44 

984 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

44 

985 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

44 

986 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

44 

987 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

44 

988 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

44 

989 

5196 

5240 

5284 

5328 

5372 

5416 

5460 

5504 

5547 

5591 

44 

990 

5635 

5679 

5723 

5767 

5811 

5854 

5898 

5942 

5986 

6030 

44 

991 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

44 

992 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

44 

993 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

44 

994 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

44 

995 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

44 

996 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

44 

997 

8695 

8739 

8782 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

44 

998 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9522 

44 

999 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

43 

N. 

O 

1 

2 

3 

4 

5 

6 

7 



S 

0 

r>. 

734  APPENDIX. 

The  Application  of  Logarithms. — The  logarithm  of  a  number  is  set  down  as  a  decimal, 
and  addition  of  ciphers  to  numbers  does  not  change  the  logarithm ;  it  is  the  same  for  11, 
110,  1100,  but  the  value  of  the  number  is  established  by  figures  to  the  left  of  the  decimal 
point ;  thus,  if  the  number  is  among  the  units,  the  characteristic  is  0 ;  if  in  the  tens,  1 ; 
in  the  hundreds,  2 ;  thousands,  3 ;  tens  of  thousands,  4,  and  so  on  ;  if  the  number  is  a 
decimal  fraction  and  the  first  figure  a  tenth,  the  characteristic  is  1,  if  hundredths  2,  thou- 
sandths 3~ 

Multiplication  of  two  numbers  is  performed  by  the  addition  of  their  logarithms  and 
characteristics,  and  finding  the  number  corresponding  to  their  sum  ;  thus,  to  multiply  119 

by  2760. 

Characteristic  of  119  2,  logarithm.  2-075547 

"  2760  3,          "  3-440909 

5-516456 

3284  403 

401  D  =  132)53(401 


328440-1  528 

200 

132 

68 

As  the  characteristic  is  5,  the  result  is  6  figures  of  whole  numbers. 
Division  is  performed  by  subtracting  the  logarithm  of  the  divisor  from  that  of  the  divi- 
dend, and  finding  the  logarithm  of  the  remainder  for  the  quotient.    But  if  the  divisor  is 
the  larger,  then  the  characteristic  of  the  remainder  is  —  . 
Thus,  to  divide  500  by  63008. 

Logarithm  of  500  2-698970 

Logarithm  of  63000  =  4-799341 


Logarithm  of  63008  4.799396 

Corresponding  number  -007985  =  3-899574 

Numbers  are  raised  to  any  power  by  multiplying  their  logarithm  by  the  exponents,  and 
roots  are  extracted  by  dividing  the  logarithm.  Thus,  to  get  the  square  of  any  number,  its 
logarithm  is  multiplied  by  2,  for  the  cube  by  3,  for  the  4th  power  by  4  ;  in  like  manner,  to 
obtain  the  square  root  of  the  number,  divide  the  logarithm  by  2  ;  by  3  for  */  ;  by  4 
for*/- 

The  roots  of  numbers  are  better  expressed  by  fractional  exponents,  thus:  V#  by  a1/a> 
tya  by  a1  8. 

The  raising  of  numbers  to  different  powers  is  extremely  simple,  by  logarithms,  when 
the  numbers  are  whole  numbers,  but  becomes  somewhat  more  complicated  when  the  num- 
bers are  decimals. 

Thus,  to  find  the  4th  power  of  -07. 

Logarithm  -07  2-845098 

_  4 

8     3-380392 

Number  -00002401  5-380392 

To  extract  the  4th  root  of  -07— 

Logarithm  -07  2-845098 

Add  2  to  the  characteristic  to  make  it 

divisible  by  4,  and  a  positive  2  to  the  2-2-845098 

logarithm  to  balance  it.  4)4'2  -845098 

Number  -5143  1-    711274 


APPENDIX.  735 

The  exponent  of  a  root  is  often  a  decimal ;  thus  the  */'07  may  be  expressed  by  *07'8B. 

Logarithm  -07  2-845098 

-25 

4225490 
1690196 
•"5-21127450 
•5-5 


Number  -5143  1-71127450 

NOTE.  —  In  this  example,  *5  is  added  to  the  resultant  characteristic  to  bring  it  to  an  integer,  and 
an  equal  positive  amount  to  the  logarithm  to  balance  it. 

The  same  logarithm  as  by  dividing  by  4-  and  corresponding  to  the  number  -5143.  The 
rule  is  to  consider  the  logarithm  as  a  plus  quantity,  and  multiply  by  the  exponent  and  the 
characteristic  as  minus,  and.  after  similar  multiplication,  subtract  it  from  the  first  product. 
When  a  characteristic  has  a  minus  sign  (3),  and  it  is  to  be  subtracted,  the  sign  is  changed 
-and  added. 

Thus,  to  divide  10-  by  TV 

Logarithm  10-  1-00000 

rV  I  _ 

Logarithm  of  100-  2-000 

To  divide  TV  Logarithm  1-00000 

b  2 


_ 
Logarithm  of  10-  1-0000 

To  divide  y^  Logarithm  3-00000 

by  100  2 

Logarithm  of  -00001  5 


INDEX. 


Acoustics  applied  to  rooms,  etc.,  530,  542 
Adcock's  table  of  teeth,  280. 
Air,  flow  of,  through  pipes,  689. 
Alphabets,  65. 

Anchors  for  floor-beams,  473. 
Animals,  forms  of,  650. 
Apartment-houses,  plans  for,  516. 
Apothecaries'  weight,  674. 
Arch  bridges,  432. 

table  of,  437. 

Arch,  Roman  cylindrical  masonry,  475. 
Arches,  574. 

ARCHITECTURAL  DRAWING,  461-601. 
Architecture,  orders  of,  Byzantine,  572. 

Composite,  571. 

Corinthian,  569. 

Domestic,  461. 

Doric,  566. 

Gothic,  572. 

Greek  and  Roman,  564. 

Ionic,  569. 

of  Houses,  461. 

Roman,  571. 

Tuscan,  566. 

Architectural  ornament,  590. 
Areas  of  circles,  691. 
Ashti  reservoir,  379. 
Asphalt  pavement,  404. 
Atkinson,  Edward,  on  use  of  ropes  in  place  of 

belts,  274. 

Automatic  valves,  335. 
Averaging  speed  of  floats  by  diagram,  72. 
Avoirdupois  weight,  674. 
Axle,  differential,  208. 
Axles,  245. 

Backing  paper  and  drawings,  58. 

Ballast  for  roads,  406. 

Balloon  frame,  469. 

Barns,  542. 

Bases,  588. 

Basilicas,  plans  of,  532. 

Bath-tubs,  498. 

'47 


Beams,  strength  of  composite,  238. 

of  iron,  230. 

of  wood,  226. 

Beam,  working,  of  engine,  322. 
Bearings  for  shafts,  245. 
Bearing,  suspension,  for  upright  shaft,  258. 
Bed-rooms,  496. 
Belgian  pavement,  403. 
Belts,  270. 

horse-power  of,  273. 

ropes  instead  of,  274,  299. 
Bends  or  angles  in  pipes,  562. 
Beton,  190. 
Bevel-wheel,  isometrical  projection  of,  630. 

projections  of,  288. 
Bismarck  Bridge  foundation,  432. 

pier,  432. 

Bituminous  cement,  190. 

Blast-pipes,  table  of  losses  of  pressure  per  100 
feet,  690. 

table  for  equalizing  diameter  of,  690. 
Blinds,  framing,  479. 
Blocks,  gin,  301. 

tackle,  301. 

Blue-print  process,  164. 
Board,  drawing,  55. 
Boiler  tubes,  weight,  etc.,  677. 
Boilers,  flue,  349. 

Hartford  Steain-Boiler  and  Inspection  Co.,  438. 

horizontal  tubular,  346. 

locomotive,  442. 

marine,  442. 

setting,  438. 

Shapley,  349. 

vertical,  350. 
Bolts  and  nuts,  239. 
Bonne's  projection,  168. 
Bonomi,  Joseph,  proportions  of  the  human  frame 

by,  643. 

Boston  Water- Works  conduit,  393. 
Box-car,  New  York  Central  and  Hudson  River 

Railroad,  453. 
Bracing,  general  principles  of,  407. 


738 


INDEX. 


Branches  in  pipes,  562. 

Brass,    Thurston's     graphic     representation    of 
strength  of,  195. 

plates,  weight  of,  676. 

rods,  weight  of,  678. 

tubes,  weight  of,  678. 

wire,  weight  of,  676. 
Bridges,  and  roofs,  407. 

arch,  432. 

Bismarck,  432. 

ferry-landing,  431. 

Howe  truss,  421. 

iron  deck  lattice-girder,  427. 

iron  plate-girder,  424. 

skew  arch,  435. 

suspension,  437. 

table  of  arch,  437. 

table  of  suspension,  438. 

truss  combination,  424. 

trusses,  rules  for,  419. 

wooden  truss,  421. 

wrought-iron  truss,  428. 
Bricks,  188. 

weight  of,  190. 
Brooklyn,  N.  Y.,  conduit  of  water-works,  392. 

sewers,  398. 

pipe-joints,  395. 
Building  materials,  182. 

artificial,  188. 
Buildings  in  the  city  of  New  York,  extracts  from 

acts  relating  to,  665. 
Burden's  rivets,  weight  of,  679. 
Buttresses,  576. 
Byzantine  church  plan,  532. 

Campaniles,  577. 
Canals.  384. 

Erie,  384. 

locks,  386. 

locks,  specifications  for  New  York  State  canals, 
388. 

Northern,  at  Lowell,  Mass.,  385. 
Capacity,  measures  of,  673. 
Capitals,  588. 

Car,  box,  New  York  Central  and  Hudson  River 
Railroad,  453. 

Pennsylvania  passenger,  453. 
Carriage-house,  542. 
Castings,  192. 

Cast-iron  columns,  strength  of,  221. 
Ceilings,  brick,  Italian,  475. 
Cement,  189. 

bituminous,  190. 
Central  Park  gravel  roads,  405. 
Center  of  gravity,  200. 
Chain  cables,  303. 


Chain  wheel,  302. 

Chains  and  ropes  of  equal  strength,  300. 

Chimney-tops,  548. 

Chimneys,  442. 

for  houses,  493. 
Church,  Gothic,  plan,  532-538. 

Romanesque,  plan,  532. 
Churches,  English,  at  Hague,  534. 

Greek,  Roman,  English,  Byzantine,   Basilica, 
532. 

London  Wesleyan,  534. 

Roman  Catholic  Cathedral,  New  York,  535. 

St.  Bartholomew,  New  York,  535. 
Circles,  properties  of,  671. 

table  of  circumference  and  areas,  691. 
Classification  of  masonry,  185. 
Closets,  497. 
Clutch  couplings,  264. 
Coals,  199. 
Coffer-dam,  365. 
Cohoes  dam,  378. 

head  gates,  380. 
Columns,  strength  of  cast-iron,  221. 

strength  of  wrought-iron,  224. 
Combination  truss-bridge,  424. 
Compound  steam-engines,  216. 

compasses,  45. 

Composite  order  of  architecture,  571. 
Concrete  floors  arched  and  groined,  474. 
Conduit  of  the  Croton  Aqueduct,  New  York,  392. 

of  the  Boston  Water-Works,  393. 

of  Nassau  Water-Works,  Brooklyn,  392. 
Conduits  for  water,  390 
Cone  pulleys,  269. 
Cones,  solidity  of,  31,  672. 
Connecting-rods,  313. 
Connections  for  rods,  312. 
Contents,  to  calculate,  144. 
Contour  lines,  152,  153,  162. 
Conventional  colors  for  topography,  171. 
Conventional  signs,  149. 

for  metals,  191. 

geological,  160. 

marine,  160. 

statistical,  162. 
Copper  plates,  weight  of,  676. 

rods,  weight  of,  678. 

tubes,  weight  of,  678. 

wire,  weight  of,  676. 
Copying  by  blue-print  process,  164. 

by  ferro-prussiate  process,  164. 

by  transfer-paper,  165. 
Copying-glass,  165. 
Corinthian  order  of  architecture,  569. 
Cornices,  588. 

plaster,  495. 


INDEX. 


739 


Counter-shaft,  269. 
Couplings,  slide  or  clutch,  264. 

for  shafts,  260. 
Cow-houses,  545. 
Cranks,  engine,  305. 

hand,  305. 
Crib,  dock,  366. 
Cross-section  paper,  157. 

uses  of,  69. 

Cross-sections,  railroad,  157. 
Croton  Aqueduct,  conduit,  392. 

dam,  375. 

new  receiving  reservoir,  394. 
Cube,  isometrical  projection  of,  625. 
Cube  roots,  table  of,  696. 
Cubes,  table  of,  696. 
Cubic  measure,  674. 
Culvert,  isometrical  projection  of,  631. 
Curbs,  403. 

Curved  lenses,  isometrical  projection  of,  629. 
Curves,  42. 
Cylinders,  solidity  of,  672. 

steam,  325. 

water,  326. 

Dam,  Ashti  Reservoir,  379. 
Dams,  374. 

Cohoes,  378. 

Croton,  375. 

head-gates  for,  380. 

Holyoke,  375. 

Lowell,  Merrimack  River,  376. 
De  Lorgne's  projection,  170. 
Design,  principles  of  architectural,  598. 
Development  of  surfaces,  104. 
Differential  screw,  208. 

axle,  208. 
Dining-rooms,  496. 
Distribution  water-works,  395. 
Dividers,  45. 
Dock,  crib,  366. 

Dome  of  brick  and  concrete,  476. 
Domes  and  vaults,  574. 
Domestic  architecture,  461. 
Doors,  sliding,  478. 

folding,  479. 

framing,  476. 
Doorways,  584. 

Doric  order  of  architecture,  566. 
DRAWING  INSTRUMENTS,  40-77. 
Drawing-pen,  exercises  with,  61. 
Drinker,  H.  S.,  method  of  timbering  tunnels,  449. 
Driven  wells,  tubes  for,  weight,  etc.,  of,  678. 
Dry  measure,  674. 
Dynamic  force,  210. 

table,  674. 


Eccentrics,  309. 

projections  of,  309. 
Elevators,  521. 

Ellipse,  to. find  the  area  or  circumference,  672. 
Embankment,  Ashti  Reservoir,  379. 
ENGINEERING  DRAWING,  362-460. 
English  churches,  532. 

Equalizing  diameter  of  blast-pipes,  table  for,  690. 
Erie  Canal,  384. 

rates  compared  with  New  York  Central  and 

Hudson  River  Railroad  by  diagram,  71. 
Evaporation  from  reservoirs,  374. 
Expansion,  table   of  mean  pressures   in  steam- 
cylinders  at  different  rates  of  expansion,  682. 

Falling  bodies,  velocity  of,  etc.,  210. 

Fanning,  J.  F.,  table  of  flow  of  water  through 

pipes,  685. 

Farm  Pond  head-gates,  Boston  Water- Works,  384. 
Ferro-prussiate  paper  for  copying,  164. 
Ferry-landing  bridge,  431. 
Figure-drawing,  human,  Bonomi,  643. 

Villard  de  Hennecourt's,  645. 
Finishing  topographical  map,  174. 
Fire-places,  492. 
Fire-proof  French  floors,  474. 

concrete  floors,  474. 
Fire-resisting  floors,  472,  474. 
Flats.     See  Apartment-houses. 
Flooring,  470. 
Floors,  brick  arch  and  iron  beams,  474. 

concrete,  arched  and  groined,  474. 

fire-resisting,  474. 

mill  fire-retarding,  472. 

single  and  double,  472. 
Flow  of  water,  683. 
Flue  boilers,  3491 
Flues,  547. 

for  houses,  492. 
Flumes,  390. 

Forces,  parallelogram  of,  208. 
Foundations,  181. 

Bismarck  Bridge,  432. 

coffer-dam,  365. 

concrete,  362. 

iron  piles,  364. 

machine,  449. 

pile,  363. 

sheet-piling,  364. 

steam-engine,  449. 

stone,  362. 

Susquehanna  Bridge,  373. 

timber,  362. 

under  water,  371. 
Frame,  balloon,  469: 

houses,  468. 


740 


INDEX. 


Frames,  355. 
Framing,  468. 

doors,  476. 

roofs,  493. 

scarfing,  lapping,  473. 

windows,  sash,  and  blinds,  479. 
Francis,  J.  B.,  formula  for  flow  over  weirs,  dia- 
gram of,  683. 
FREE-HAND  DRAWING,  639-664. 

drawing,  elementary  exercises  in,  639. 
Freight  shed,  wood,  419. 
French  flats.     See  Apartment-houses. 
Friction,  211. 

Morin's  experiments  on,  212. 
Frictional  gearing,  297. 

Fteley,  A.,  formula  for  flow  through  sewers,  685. 
Furnaces,  hot  air,  549. 

Galvanized  iron,  spiral  riveted  pipes,  weight,  etc., 

678. 
Gas,  for  lighting,  564. 

flow  of,  through  cast-iron  mains,  689. 

supply,  401. 
Gearing.  275. 

frictional,  297. 

mortise-wheels,  283. 

projections  of  bevel-wheels,  288. 

projections  of  spur-wheel,  284. 

teeth  of,  277. 

wedge,  299. 

worm,  296. 

Geometrical  definitions,  2. 
GEOMETRICAL  PROBLEMS,  CONSTRUCTION  OF,  1-39. 
Gin-blocks,  301. 
Glass,  197. 

sizes  of  cylinder  and  plate,  482. 
Globular  or  equidistant  projection  of  the  sphere, 

166. 

Glue,  mouth,  57. 
Gothic  architecture,  572. 
Gothic  church-plan,  532-538. 
Grade  of  roads  (table),  405. 
Graphic  diagrams — belts,  the  power  of,  273. 

charges  for  transport  of  merchandise  on  rail- 
road and  canal,  71. 

crank  eyes,  306. 

movements  of  a  float  in  a  canal,  72. 

rainfall,  temperature,  and  mortality,  74. 

speed  and  resistance  of  railway -trains,  73. 

steam-expansion  in  single  and  compound  cylin- 
ders, 215. 

strength  of  wrought-iron  columns,  225. 

strength  of  wrought-iron  girders,  237. 

strength  of  wrought-iron  shafts,  248. 

-teeth  of  wheels,  277. 

Thurston's  strength  of  alloys,  19(5. 


Graphic  diagrams — time-table  of  railroad,  72. 

water-flow  through  pipes,  686. 

water-flow  through  sewers,  688. 

weights  and  measures,  76. 
Gravel-roads  in  Central  Park,  405. 
Gravity,  center  of,  200. 
Greek  architecture,  orders  of,  564. 
Greek  churches,  532. 
Greenhouses,  546. 
Groin,  Roman,  475. 
Gutters,  forms  of,  493. 

Halls,  music,  541 ;  legislative,  541. 
Hand-valves,  337. 
Hangers,  254. 

Seller's,  260. 
Head-gates,  380. 

Cohoes  dam,  380. 

Farm  Pond,  Boston  Water- Works,  384. 

made  at  Holyoke,  Mass.,  390. 
Heating,  methods  of,  549. 

open  fires,  549. 

steam  and  hot  water,  551. 

stoves,  549  ;  hot-air  furnaces,  549. 
Helix,  102. 

Hills,  Von  Eggloffstein's  system  of  representing, 
154. 

representation  of,  152. 
Hoists,  power,  521. 
Holyoke  dam,  375. 
Hoofs  of  animals,  651. 
Hooks,  form  of,  303. 

Hoosac  Tunnel,  method  of  timbering,  453. 
Horses,  movements  of,  651. 
Horse-power,  etc.,  213. 

of  belts,  273. 
Hospitals,  542. 

Hot-water  heating  apparatus,  551. 
Houses,  architecture  of,  461. 

elevation  of  high-stoop  houses,  513. 

frame,  468. 

plans  for  apartment,  516. 

plans  and  elevations  of  country  residences,  509. 

plans  and  elevations  of,  in  Queen  Anne  style, 
503. 

plans  for  rooms  in,  498. 

plans  of  tenement,  513. 
Howe  truss-bridges,  421. 
Human  frame,  proportions  of,  by  Joseph  Bonomi, 

643. 

Hydrants,  341. 
Hydraulic  press,  210. 
Hydrometrical  surveys,  159. 

Illuminating-tile,  516. 

Inches  in  decimals  of  a  foot,  673. 


INDEX. 


741 


Inclined  forces,  206. 

plane,  205. 
Indicator  cards,  216. 
Ink,  China,  60. 

Instruments,  management  of,  59. 
Ionic  order  of  architecture,  569. 
Iron,  weight  of  rolled,  676. 

weight  of  wrought  plates,  676. 

wire,  weight  of,  676. 

ISOMETRICAL    DRAWING,  625-638. 

Isometrical  projection,  633. 

Joinings  of  timber,  472. 
Joints  of  Brooklyn  pipes,  395. 

riveted,  342. 
Joists,  size  of,  471. 
Journals,  245-251. 

Keys,  249. 

Land-plans,  railroad,  158. 

Lands,  division  of  U.  S.,  147. 

Lanza  on  strength  of  wooden  posts,  221. 

Lapping,  timber,  473. 

Latitudes  and  departures,  table  of,  704. 

Lead  pipe,  weight  of,  680. 

plates,  weight  of,  676. 
Lecture-rooms,  541. 
Legislative  halls,  541. 

Lehmann'a  system  of  representing  slopes,  153. 
Lettering  for  maps,  174. 
Letters,  samples  of,  65. 
Levers,  202. 

form  of  hand,  304. 

form  of  foot,  304. 
Lineal  measure,  672. 
Liquid  measure,  673. 
Locks  of  canals,  386. 
Locomotive  boilers,  442. 
Logarithms,  application  and  use  of,  734. 

of  numbers,  table  of,  71$. 
Lowell  dam,  Merrimack  River,  376. 

water-power,  214. 

Macadam  roads,  404. 

Machine  and  blacksmith  shop,  perspective  view, 
etc.,  521. 

MACHINE  DESIGN    AND     MECHANICAL   CONSTRUC- 
TIONS, 220-361. 

Machine-foundations,  449. 

Machines,  location  of,  444. 

Man-holes  for  sewers  in  New  York,  399. 

Mantel-piece,  492. 

Map,  finishing  topographical,  174. 
projections,  165. 

Maps,  lettering,  174. 


Maps,  railway,  156. 

titles,  176. 

transferring,  162. 

United  States  Coast  Survey,  table  for  project- 
ing maps,  168. 
Marine  boilers,  442. 
Marine  surveys,  159. 
Masonry,  classification  of,  185. 

technical  terms  of,  185. 
MATERIALS,  181-199. 
Materials,  building,  182. 

bricks,  189. 

coal,  flame,  steam,  199. 

glass,  rubber,  198. 

metals,  197. 

mortars  and  cements,  191. 

stone,  188. 

wood,  185. 

Mechanical  work  or  effect,  212. 
MECHANICS,  200-219. 
Melting-point  of  metals,  195. 
Mensuration,  67  i. 
Mercator's  projection,  171  . 
Meridional  parts,  table  of,  171. 
Metals,  191. 

conventional  signs,  191. 

crushing  strength,  195. 

melting-point,  195. 

specific  gravity,  195. 

tensile  strength,  195. 

weight,  175. 

Metric  system,  diagram  of  equivalent  values,  70. 
Moldings,  586,  589,  594. 

Greek  and  Roman  names  and  forms,  483. 
Morin's  experiments  on  friction,  212. 
Mortality  shown  by  diagram,  73. 
Mortars,  189. 
Mortise-wheels,  283. 
Mounting  paper  and  drawings,  58. 
Music-halls,  541. 

Nails,  weight  and  length,  680. 
Nature,  drawings  from,  directions  for,  653. 
New  York  building  laws,  extracts  from,  665. 
New  York  Central  and  Hudson  River  Railroad 

rates  compared  with  Erie  Canal  by  diagram, 

71. 
New  York  Central  and  Hudson  River  Railroad 

box-car,  453. 

New  York  city  sewer  catch-basins,  400. 
sewer  man-holes,  399. 
streets,  widths,  etc.,  402. 
New  York  docks,  bulkhead- walls,  365. 
New  York,  New  Haven  and  Hartford  Railroad. 

diagram  of  time-table,  71. 
Northern  Canal,  Lowell,  Mass.,  385. 


742 


INDEX. 


Noses  of  animals,  653. 

Organs,  535. 

Ornament,  architectural,  590. 
Ornaments  of  the  Renaissance,  596. 
ORTHOGRAPHIC  PROJECTION,  78-1U9. 

Paints,  198. 
Pantagraph,  54. 
Pantries,  497. 
Paper,  backing,  58. 

cross-section,  profile,  157. 

fixing  down,  57. 

mounting,  58. 

stretching,  57. 

uses  of  cross-section,  69. 

varieties  and  sizes  of,  56. 
Parabola,  area  of,  612. 
Parallel  motion,  324. 

ruler,  42. 

Parallelogram  of  forces,  208. 
Parallelepipeds,  solidity  of,  672. 
Parapets,  594. 
Paris  streets,  402. 
Parlors,  496. 
Partitions,  469. 
Passage-ways  in  houses,  497. 
Patent-Office  drawings,  directions  for,  670. 
Pavement,  asphalt,  404. 

Belgian,  403. 

wooden,  404. 
Paws  of  animals,  651. 
Pen,  drawing,  43. 

exercises  with  drawing,  61. 
Penetrations  or  Intersections  of  solids,  90. 
Pennsylvania  passenger-car,  453. 
PERSPECTIVE  DRAWING,  602-624. 
Perspective,  angular,  612. 

parallel,  604. 

scale  for  drawing,  608. 
Pew,  size  of,  etc.,  531,  535.    ' 
Piers  of  stone,  431. 

stone,  Bismarck  Bridge,  432. 

wooden  pile,  431. 

wrought  iron,  432. 
Pile  foundation?,  363. 

piers,  431. 
Piles,  iron,  364. 
Piling,  shoot,  364. 
Pillow-block,  252. 

isometrical  projection  of,  631. 
Pinion  and  rack,  291. 
Pipe-connections.  350. 
Pipe,  lead,  weight  of,  680. 

joints  of  Brooklyn,  395. 
Pipes,  diagram  of  flow  of  water  through,  685. 


Pipes,   galvanized  iron,   spiral,  riveted,  weight, 
etc.,  678. 

table  for   equalizing  the   diameter   of  blast- 
pipe,  690. 

table  of  losses  of,  pressure  per  100  feet  in 

blast-pipes,  690. 
Pistons,  327. 
Plastering,  190,  494. 
PLOTTING,  137-148. 
Plumber  block,  252. 
Plumbing,  555. 

water-supply,  555-563. 
Polyconnic  projection,  168. 
Posts,  strength  of  wooden,  221. 
Power,  horse,  213. 

steam,  214. 

water,  214. 
Press,  hydraulic,  210. 

Pressure,  table  of  loss  of,  per  100  feet  in  blast- 
pipes,  690. 

Prisms,  solidity  of,  672. 
Privies,  497. 
Profiles,  railroad,  157. 
Profile  paper,  157. 
PROJECTION,  ORTHOGRAPHIC,  78-109. 

Bonne's,  168. 

De  Lorgne's,  170. 

globular  or  equidistant,  166. 

Mercator's,  171. 

polyconic,  168. 

stereographic,  167. 
Projections  of  simple  bodies,  81. 

for  maps,  165. 
Protractor,  53. 
Pulley,  204. 
Pulleys,  266. 

cone,  269. 

speed  of,  266. 
Pyramids,  solidity  of,  672. 

Queen  Anne  style,  plans  and  elevations  for  house 
of,  503. 

Rack  and  pinion,  293. 
Railroad  cross-sections,  157. 

land  plans,  158. 

profiles,  157. 
Railroads,  ballast  for,  406. 

sections  of  rail,  406. 
Rails,  sections  for  railroads,  406. 
Railway  maps,  156. 

stock,  453. 

Reservoir,  new  Croton  Receiving,  394. 
Reservoirs,  Ashti,  379. 

receiving,  394. 
Retaining  walls,  365. 


INDEX. 


743 


Retaining  walls,  crib  docks,  366. 

New  York  docks,  365. 

Thames  embarkment,  369. 
Riveted  joints,  342. 
Rivets,  Burden's,  weight  of,  679. 
Roads,  402. 

ballast  for,  406. 

Central  Park  gravel,  405. 

Macadam,  404. 

table  of  grades,  405. 
Robertson's   grooved-surface   frictional   gearing, 

299. 

Rolled  iron,  table  of  weight  of,  675. 
Romanesque  church-plan,  532. 
Roman  orders  of  architecture,  564,  571. 

churches,  532. 

cylindrical  masonry  arch,  475. 

groin,  475. 

Roofs,  Gothic  church,  538. 
Roof-truss,  isometrical  projection  of,  631. 
Roofs  and  bridges,  407. 

bracing  for  wooden,  409. 

framing,  forms  of,  493. 

of  iron,  414. 

wooden  freight-shed,  419. 
Rooms   and   passages,   sizes,   arrangement,   and 

proportions  of,  495. 
Ropes  and  chains  of  equal  strength,  300. 

strength  of,  301. 

used  as  belts,  274,  299. 
Rubber,  198. 
Rulers,  40. 

Russell,  J.  Scott,  wave-line  principle  of  ship-con- 
struction, 458. 

Safety-valves,  341. 
Sash,  framing,  479. 

.Saunders's  experiments  on  sound,  530. 
Scale  of  perspective  drawing,  608. 

guard,  49. 
Scales,  47. 

Scarfing,  timber,  473. 

School-house,  isometrical  projection  of,  631. 
School-houses,  ventilation  and  light,  530. 

plans     and     elevation    of     New    York     city, 
527. 

plans  and  elevation  of  Cleveland  city,  527. 

plans,  elevations,  etc.,  521. 
Screws,  205,  241,  294. 
Screw,  wheel  and  endless,  296. 

differential,  208. 

Seats  in  general,  space  occupied  by,  531. 
Sewers,  398. 

catch-basins  and  man-holes  of  New  York,  399, 
400. 

diagrams  and  formula  of  flow  through,  685. 


Sewers,  isometrical  projection  of,  in  Thames  Em- 
bankment, 631. 

large  street,  Washington,  D.  C.,  399. 

of  Brooklyn,  N.  Y.,  398. 

overflow  and  outlet  of  the  Victoria  and  Re- 
gent Streets  sewers,  Thames  Embankment, 
371. 

Sewer  pipe-connections,  555. 
SHADES  AND  SHADOWS,  110-136. 
Shade-lines,  107. 
Shading  and  shadows,  manipulation  of,  126. 

elaboration  of,  129. 
Shaft,  counter,  269. 
Shafting,  249. 
Shafts,  245. 

couplings  for,  260. 

upright,  256. 
Shapley  boiler,  349. 
Shearing  stress,  225. 
Sheet-piling,  364. 
Ship-construction,  wave-line  principle  by  J.  Scott 

Russell,  458. 
Sidewalks,  402. 

Sines  and  cosines,  table  of  natural,  710. 
Sinks,  557. 

Skew-arch  bridges,  435. 
Slide  couplings,  264. 

Slopes,  United  States  Coast  Survey  system  of  rep- 
resenting, 153. 

Lehmann's  system  of  representing,  153. 
Solid  measure,  674. 
Soil-pipe  connections,  555. 
Sound,  Saunders's  experiments  on,  530. 
Specials,  water-pipe,  396. 
Specific  gravity  of  metals,  195. 
Sphere,  globular  or  equidistant  projection  of,  166. 

area  of,  672. 

solidity  of,  672. 

Spikes,  wrought,  weight  of,  679. 
Spires,  578. 
Spur-wheel  (internal)  driving  a  pinion,  293. 

driven  by  a  pinion,  294. 

projections  of,  284. 
Square  roots,  table  of,  696. 
Squares,  T,  41. 

table  of,  696. 
Stables,  542. 
Stairs,  iron,  490. 

framing,  etc.,  485. 
Stalls  for  horses,  543. 
Standard,  253. 
Static  force,  200. 
Steam-cylinders,  325. 

table  of  mean  pressures  in,  at  different  rates  of 

expansion,  682. 
Steam-engine,  214. 


INDEX. 


Steam-engine,  compound,  216. 

foundation,  449. 

indicator  cards,  216. 
Steam-heating  apparatus,  551. 
Steam-power,  214. 

Steam,  table  of  propei'ties  of  saturated,  681. 
Step  for  upright  shaft,  256. 
Stereographic  projection,  167. 
Stones,  185. 

varieties  of,  187. 

weight  of,  191. 
Streams,  flow  of,  374. 
Strength  of  brass  alloy,  195. 

of  cast-iron  columns,  221. 

of  composite  beams,  238. 

of  iron  beams,  230. 

of  metals,  195. 

of  wooden  beams,  226. 

of  wooden  posts,  221. 

of  wrought-iron  columns,  224. 
Stores,  plans  and  elevations  of,  516. 
Stories,  height  of,  in  houses,  497. 

height  of,  in  stores,  516. 
String  courses,  588. 
Stoves,  549. 
Streets,  asphalt  pavement,  404. 

Belgian  pavement,  403. 

carnage-  way,  403. 

curbs,  403. 

Macadam,  404. 

of  New  York,  widths,  etc.,  402. 

of  Paris,  402. 

sidewalks,  402. 

wooden  pavement,  404. 
Stress,  220. 

shearing,  225. 

torsional,  225. 

transverse,  226. 
Stuffing-boxes,  330. 
Sulphur,  196. 
Sunday-school  room,  535. 
Surface,  measures  of,  673. 
Surveys,  hydrometrical,  159. 

marine,  159. 
Suspension-bridges,  437. 

table  of,  438. 

Susquehanna  Bridge  foundations,  373. 
curves,  42. 


Table  for  projecting  maps,  168. 

of  meridional  parts,  171. 

traverse.     See  Appendix. 
Tables  of  areas  of  circles,  691. 

blast-pipes,  equivalent  areas  of,  690. 

blast-pipes,  losses  of  pressure  in,  690. 

boilers,  number  of  tubes,  346. 


Tables  of  boilers,  stay-bolts,  347. 

boilers,  weight  of  tubes,  677. 

bolts  and  nuts,  244. 

brass  plates,  tubes,  rods,  wire,  weight  of,  676. 

bridges,  arch,  432. 

bridges,  suspension,  438. 

chains  and  ropes,  equivalent  strength,  300. 

circles,  circumferences  and  areas,  691. 

copper  plates,  tubes,  rod,  wire,  weight  of,  676. 

cubes  and  cube  roots,  696. 

expansion,  mean  pressures  at  different  rates  of, 
cut  off,  682. 

gas-pipes,  weight  of,  402. 

gears,  teeth  of,  280. 

hooks,  proportions  of,  304. 

iron  angle  and  channel,  235. 

iron  plates,  tubes,  rods,  wire,  weight  of,  675. 

iron,  safe  load  of  cast-iron  columns,  222. 

iron,  safe  load  of  wrought-iron  columns,  223. 

iron,  safe  load  of  I  beams,  233. 

iron  tubes  and  couplings,  sizes  of,  352. 

journals,  dimensions  of,  245. 

latitudes  and  departures,  704. 

lead  in  joints  of  pipes,  396. 

lead  pipes,  sizes  and  weights,  680. 

logarithms,  719. 

maps  for  meridional  parts,  171. 

maps  for  projections  of,  168. 

metals  and  alloys,  weight  and  strength,  195. 

nails,  weight  of,  680. 

rivets,  pitch  of,  343. 

rivets,  weight  of,  679. 

sheaves,  sizes  of,  301. 

sines  and  cosines,  natural,  710. 

spikes,  weight  of,  679. 

squares  and  square  roots,  696. 

steam,  properties  of,  681. 

theatres,  dimensions  of,  541. 

valves,  dimensions  of,  336,  339. 

water-discharge  over  weirs,  684. 

water-pipes,  dimensions  of,  396. 

water,  weight  of,  680. 

weights  and  measures,  672. 

wooden  beams,  safe  loads,  229. 

working-beams  of  engines,  324. 
Tackle-blocks,  301. 
Teeth  of  gearing,  277. 

Adcock's  table  of,  280. 
Tenement-houses,  plans  of,  513. 
Thames  Embankment,  river  wall,  369. 
Theatres,  plans,  539. 

Ferguson's  plan,  540. 

Wagner's,  540. 

table  of  dimensions,  541. 

Thurston's  graphic  representation  of  strength  of 
brass  alloys,  195. 


INDEX. 


745 


Tile,  illuminating,  516. 
Tinting,  methods  of,  126. 
Titles  for  maps,  176. 
Topographical  map-finishing,  174. 
TOPOGRAPHICAL  DRAWING,  149-180. 
Topography,  colored,  171. 

conventional  colors,  171. 
Torsional  stress,  225. 
Towers,  577,  580. 
Transfer-paper  for  copying,  165. 
Transferring  maps,  162. 
Transverse  stress,  226. 
Traps  in  pipes,  562. 
Traverse-table,  704,  710. 

use  of,  in  plotting,  etc.,  142. 
Triangle  and  square,  use  of,  33. 
Triangles,  40. 

properties  of,  671. 
Troy  weight,  674. 

Truss,  isometrical  projection  of  roof,  631. 
Trusses,  bridge,  rules  for,  419. 
Tubes  for  boiler,  weight  of,  677. 

driven  wells,  weight  of,  678. 

weight,  etc.,  of  wrought-iron  welded,  677. 
Tubular  boilers,  346. 
Tunnels,  method  of  working,  449. 
Tunnel,  Hoosac,  method  of  timbering,  453. 
Tuscan  order  of  architecture,  566. 
Type,  samples  of,  65. 

Upright  shafts,  256. 

Urinals,  563. 

United  States  Coast  Survey  tables  for  projecting 

maps,  168. 

system  of  representing  slopes,  153. 
United  States  lands,  division  of,  etc.,  147. 

Valves,  automatic,  335. 

hand,  337. 

safety,  341. 

steam  cylinder,  331. 
Varnishing  drawings,  etc.,  58. 
Vaulting,  Greek  and  Roman,  574. 

Gothic,  575. 
Vaults  and  domes,  574. 
Velocity  of  falling  bodies,  210. 
Ventilation  and  warming  in  general,  547. 
Vestry-rooms,  535. 
Villard  de  Hennecourt's  design  of  human  figures, 

645. 

Von  Eggloff stem's  system  of  representing  hills, 
154. 

Walls,  retaining,  365. 

brick,  stone,  concrete,  467. 
bulkhead,  of  New  York  docks,  365. 


Walls,  construction  of,  461. 

of  Northern  Canal,  at  Lowell,  Mass.,  385. 

wooden,  468. 
Washers,  244. 

Washington,  D.  C.,  largest  sewer,  399. 
Wash-tubs,  558. 
Water-closets,  497,  559-563. 
Water  cylinders,  326. 
Water,  flow  of,  683. 

flow  through  pipes,  685. 

power,  214. 

power  at  Lowell,  214. 

supply,  555. 

table   of    discharge   of    weir   one   foot   long, 
684. 

weight  of,  680. 

works  distribution,  395. 

works  distribution,  house  services,  397. 

works  distribution,  specials,  396. 

works  distribution,  specifications  for  Brooklyn 

pipe,  397. 
Wave-line    principle  of  ship-construction,  by  J. 

Scott  Russell,  458. 

Weaving-room,  location  of  machinery,  444. 
Wedge,  205. 

gearing,  299. 

solidity  of,  672. 
Weights  and  measures,  672. 
Weight,  comparison  of,  674. 

of  brick,  190. 

of  materials,  182,  191. 

of  metals,  195. 

of  stones,  191. 

of  woods,  185. 

Weir,  table  of  discharge  of,  one  foot  long,  684. 
Wheel  and  axle,  203. 

and  endless  screw,  296. 

isometrical  projection  of  bevel,  630. 
Windows,  various  forms  of,  581. 

dormer,  479. 

sash  and  blinds,  479. 
Wooden  posts,  Lanza  on  the  strength  of,  221. 

pavement,  404. 
Woods,  weight  of,  185. 

characteristics  and  use  of,  183. 

representation  of,  182. 
Working-beam  of  engine,  322. 
Worm-gearing,  296. 
Wrought-iroii  columns,  222. 

plates,  weight  of,  676. 

strength  of,  224. 

welded  tubes,  weight,  etc.,  of,  677. 

Yoke-hanger,  254. 

Zinc  plates,  weight  of,  676. 


: 


PL.  I. 


T  VI 


PL.  II. 


a 


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Fig.  7. 


Fig.  5. 


PL.  EL 


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PL.I\: 


riG.2. 


FIG.8 


FI6.6. 


FI6.5. 


PL.V: 


T//< 


PL 


PL 


I. 


S  TAT  EX   ISLAND 

CONTOURS  20  TT.  APART. 

-:- 


[X. 


GEOLOGICAL  MAP  OF 

NEW  JERSEY 


GEORGE. H.COOK,STATE  GEOLOGIST 


&P' 


FORMATIONS 

CONGLOMERATES 

SHALES 
TRAP 

RED   SANDSTONE 
ANDTRAP  ROCKS 


GRAVELLY  EART 
GLASS  SAND  AND 


SHALES.  ROOFING  SLATES 

AND 
SLATY  SANDSTONES 


SANDY  CLAYS 
UPPER  MARL  BED 


CAUDA  GALLI  CRIT 
ORISKANY  SANDSTONE 


MIDW-E  MARL  BED 
RED/SAND  RED 
LOWER  MARLBED 
LAMINATED  SAND 


FOSSILIFEROUS  LIMESTE 
MAGNESIAN    LIMESTONE 


LR  HELDERBERGLIME 
STONE  AND  WAT- LIME 
RED  SLATESANDSAND 

STONE 

SANDSTONE  AND 
GONG. OF  KITTATINNYMT 


E.5LATYBRI 
AND  GREEN  POND  MT 

CONGLOMERATE 


AND  CLAY  MARLS 
POTTERS, AND  FIRE 
CLAYS  AND  SANDS 


GRISTALLINE  LIMESTONE 


PLJvL 


OUTLET 


PLOT. 


'L.XDI. 


PL  XV. 


PL.  XVI. 


PL  XVII. 


PL  XVIII. 


'     PL  XIX. 


PL.  XX. 


SCRAPS. 


IT  has  been  my  practice  for  many  years  to  collect,  from  the  circulars  of 
mechanics  and  their  agents,  and  from  illustrated  newspapers  and  magazines, 
varied  illustrations  of  tools  and  machines,  engineering  structures,  buildings, 
etc.,  and  arrange  them  under  their  appropriate  heads  in  scrap-books.  *They 
have  been  found  very  useful  in  assisting  me  in  designs,  not  only  enabling  me 
the  more  readily  to  make  drawings,  but  to  convey  to  the  draughtsman  the 
character  and  proportions  of  the  design  which  I  wish  to  have  made.  And  those 
parts  which  are  of  common  use  and  purchasable  in  the  market  can  be  readily 
arranged  in  position  and  executed  more  economically  than  from  a  new  design. 
There  is  a  saving  in  the  matter  of  drawing,  and  a  saving  in  the  cost  of  con- 
struction. 

By  a  proper  combination  and  arrangement  of  parts  which  have  practically 
served  a  purpose,  a  more  satisfactory  design  can  be  made  than  from  attempts 
at  originality.  Knowledge  of  what  has  been  done  is  economy  in  all  labor. 
When  the  thing  itself  can  not  be  seen  bodily,  its  picture  can  supply  its  place, 
and  its  details  can  be  studied  at  leisure ;  and,  as  the  education  of  the  eye  is 
of  essential  importance  to  the  draughtsman,  let  him  see  as  much  as  he  can 
practically,  but  yet  acquire  a  good  collection  of  scraps  from  which  to  design. 
There  are  few  constructions  from  which  something  of  education  can  not  be 
drawn,  parts  if  not  a  whole. 

In  this  view  a  small  collection  of  scraps  has  been  made  pertinent  to  the 
book.  Its  page  does  not  admit  of  the  sizes  which  will  be  found  in  the  illustrated 
papers  and  magazines — the  quarto  will  be  found  much  more  generally  useful — . 
and  a  library  of  such  scrap-books  will  furnish  material  for  a  draughtsman  which 
can  not  be  found  in  any  encyclopaedia. 

48 


SCRAPS. 


SCRAPS. 


SCRAPS. 


Hydraulic,  Stop-Valve. 


Hydraulic  Release-  Valve. 


Lever- Gate. 


Elliptic  Spring. 


Half  Elliptic  Spring. 


Vose  Graduated  Spring. 


Volute  Spring. 


Oval  Bar  Spring 


SCRAPS. 


Compound  Steam  Cylinders.     H.  M.  S.  Spartan. 


Wrought-Iron  Plates  and  Covers. 


Compressed-Air  Locomotive,  St.  Oothard  Tunnel. 


SCRAPS. 


Three- Throw  Crank. 


Forged  weight,  24  tons  11  cwt. 
Finished    "      15    "     8    " 


Weight,  25  tons  10  cwt. 


SCRAPS. 


Screw  Propeller. 
Vessel,  1400  gross  tons.     Engines,  130  nominal  English  horse-power. 


SCRAPS. 


TURBINES. 


CENTRAL   DISCHARGE. 


Turbine  with  Horizontal  Shaft. 
Longest  draft  tube,  at  Manchester,  N.  H.    26  feet  from  center  of  shaft  to  tail  water.    Fall,  40  feet. 


SCRAPS. 


SCRAPS. 


AMERICAN  LOCOMOTIVES. 


The  Consolidation. 


The  Mogul. 


Twelve- Wheeler,  Central  Pacific  Railroad. 


SCRAPS. 


ss 

I 


SCRAPS. 


Third  Avenue  Elevated  Railroad. 


Curve  at  Eighth  Avenue. 


SCRAPS. 
BUILDERS'   HARDWARE. 


Mortise-Lock,  cover  off. 


Front 

Boxed  Strike,  Front.  Sli&ing-door 

Loch. 


Thumb-Piece. 


Knob  and  Rose. 


Escutcheons. 


SCRAPS. 


Sash-Lifts. 


Hook  and  Eye. 


Shutter- 
Knob. 


SCKAPS. 


Examples  of  Ancient  Hinges  and  Doors. 


Balusters. 


Cast-Iron  Tread. 


SCRAPS. 


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Plan,  Section,  and  Elevation  of  a  Wooden  Mantel  and  Fire-Place. 


SCRAPS. 


Examples  of  Inlaid  Floors  or  Marquetry. 


49 


SCRAPS. 


SCRAPS. 


' 

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, 

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, 

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2-  J 

SCRAPS. 


Enameled  Tile. 


Terra  Coiia. 


SCRAPS. 


SOKAPS. 


SCKAPS. 


SCRAPS, 


SCRAPS. 


SCRAPS. 


SCRAPS. 


SCRAPS. 


SCRAPS. 


SCRAPS. 


SCRAPS. 


SCRAPS. 


SCRAPS, 


SCRAPS. 


SCRAPS. 


Central  Park,  New  York  City. 


SCRAPS. 


Coney  Island. 


SCRAPS. 


Corny  Island. 


<L 

*Hh 


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